ExampleConstruction: AI-generatedVerification: 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.

A concrete calculation of $(L_{1} L_{2}) L_{3} = L_{1} (L_{2} L_{3})$

Example

Let $L_{1} = {a}$ , $L_{2} = {ε, b}$ , and $L_{3} = {a, b b}$ over the alphabet ${a, b}$ . Then $(L_{1} L_{2}) L_{3} = L_{1} (L_{2} L_{3}) = {a a, a b b, a b a, a b b b} .$

Facts & Assumptions

Given: $L_{1} = {a}$ , $L_{2} = {ε, b}$ , and $L_{3} = {a, b b}$ .

[L1]

Language concatenation is $L K = {u v : u \in L and v \in K}$ by Language concatenation, powers, and Kleene star.

[L2]

Language concatenation is associative by Language concatenation is associative.

Verification

technique · direct

1.1

Using [L1], we have $L_{1} L_{2} = {a ε, a b} = {a, a b}$ . Therefore $(L_{1} L_{2}) L_{3} = {a a, a b b, a b a, a b b b} .$

givenL1

1.2

Again by [L1], $L_{2} L_{3} = {ε a, ε b b, b a, b b b} = {a, b b, b a, b b b}$ . Therefore $L_{1} (L_{2} L_{3}) = {a a, a b b, a b a, a b b b} .$

givenL1

2.1

The two explicit calculations in steps 1.1 and 1.2 agree, so this example realizes the identity of [L2] as the four-word set ${a a, a b b, a b a, a b b b}$ .

L2step 1.1step 1.2∎

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

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

A concrete calculation of (L1L2)L3=L1(L2L3)(L1​L2​)L3​=L1​(L2​L3​)