Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 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.

PDAs recognize exactly the context-free languages

Statement

A language over a finite alphabet is context-free if and only if some PDA recognizes it.

Facts & Assumptions

Given: A language L over a finite alphabet Σ.

[L1]

By Every context-free grammar has an equivalent PDA, every context-free grammar has an equivalent PDA.

[L2]

By Every PDA has an equivalent context-free grammar, every PDA has an equivalent context-free grammar.

Proof

technique · direct
1.1

If L is context-free, choose a grammar generating L. Then [L1] gives a PDA recognizing the same language.

L1given
1.2

If some PDA recognizes L, then [L2] gives a context-free grammar generating that same language.

L2given
2.1

Therefore L is context-free if and only if some PDA recognizes it.

step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

9 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