TheoremStatement: AI-adaptedProof: 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.

Decidable languages are closed under the Boolean operations

Statement

If $L, K \subseteq Σ^{*}$ are decidable, then their complement, union, intersection, difference, and symmetric difference are decidable.

Facts & Assumptions

Given: Decidable languages $L, K \subseteq Σ^{*}$ .

[L1]

By Decidable and recognizable languages, a decider halts on every input with the correct membership answer.

[L2]

By Boolean operations on languages over a fixed alphabet, complements and differences are taken inside the fixed ambient $Σ^{*}$ .

[A1]

The set identities $L ∖ K = L \cap K^{c}, L △ K = (L ∖ K) \cup (K ∖ L)$ hold for all subsets of one set.

Proof

technique · direct

1.1

If $M_{L}$ decides $L$ , then flipping its final yes-or-no answer gives a halting decider for $L^{c}$ . If $M_{L}$ and $M_{K}$ decide $L$ and $K$ , run both and combine their two halting answers by the ordinary Boolean truth tables to decide $L \cup K$ and $L \cap K$ .

L1L2given

2.1

By [A1], the same deciders decide $L ∖ K$ as $L \cap K^{c}$ and decide $L △ K$ as $(L ∖ K) \cup (K ∖ L)$ . Because each constituent operation from step 1.1 preserves halting, these languages are decidable as well.

A1step 1.1

3.1

Therefore decidable languages are closed under all five stated Boolean operations.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

John Watrous, Introduction to the Theory of Computing, Lecture 18: Further discussion of computability (standard reference, not scraped)
Jean Gallier and Jocelyn Quaintance, Introduction to the Theory of Computation: Some Notes for CIS511 (standard reference, not scraped)