ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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.

Removing one nongenerating variable and one unreachable variable

Example

Consider the grammar

S \to A B ∣ a, A \to a, B \to b, C \to c C, D \to d .

The variable $C$ is nongenerating, and $D$ is generating but unreachable from $S$ . Deleting both yields the smaller grammar

S \to A B ∣ a, A \to a, B \to b,

which generates the same language ${a b, a}$ .

Facts & Assumptions

Given: The grammar displayed above.

[L1]

Eliminating useless symbols preserves the generated language, by Eliminating useless symbols preserves the generated language.

Verification

technique · direct

1.1

The variable $C$ is nongenerating because its only production is $C \to c C$ , which never removes the variable $C$ . The variable $D$ is reachable from no sentential form derived from $S$ , because no production from $S$ , $A$ , or $B$ mentions $D$ .

given

2.1

Every terminal derivation from $S$ therefore uses only the variables $S$ , $A$ , and $B$ , and those derivations yield either $a$ or $a b$ . So deleting $C$ and $D$ leaves the language unchanged.

step 1.1

3.1

This is exactly the useless-symbol elimination guaranteed by [L1].

L1step 2.1∎

Depends on

Eliminating useless symbols preserves the generated language

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Jean Gallier and Jocelyn Quaintance, Introduction to the Theory of Computation: Some Notes for CIS511 (standard reference, not scraped)