Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-12
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.

Regular-expression denotation is structurally well-defined

Statement

For every regular expression R over an alphabet Σ, there exists a unique language L(R)⊆Σ∗ satisfying the recursive clauses of The language denoted by a regular expression.

Facts & Assumptions

Given: A regular expression R∈Reg⁡(Σ).

[L1]

By Regular expression syntax over an alphabet, every regular expression is obtained from the base symbols ∅,ε,a by finitely many applications of union, concatenation, and star.

[L2]

By The language denoted by a regular expression, the intended denotation is fixed on the three base symbols and, for a composite expression, is obtained from the denotations of its immediate subexpressions by one of the operations L∪K, LK, or L∗.

Proof

technique · direct
1.1L1given

By [L1], every regular expression has a finite construction tree, so there is a well-founded induction on the number of constructor occurrences in that tree.

2.1L1L2step 1.1induction

In the base cases R=∅, R=ε, and R=a, the language is uniquely fixed by [L2]. If R is composite, then its outermost constructor is uniquely one of +, concatenation, or ∗ by [L1]. The induction hypothesis gives unique denotations to the immediate subexpressions, and then [L2] applies one fixed language operation to those already determined languages.

3.1step 2.1∎

Therefore exactly one language L(R) satisfies the recursive clauses for the given expression R.

Depends on

Used by

Dependency tree · two levels

6 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