CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-09-12

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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.

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The encoding $w \mapsto 1 w$ is effective but not prefix-free

Statement refuted

The statement "every effective binary encoding is prefix-free" is false.

Facts & Assumptions

Given: The encoding $e (w) = 1 w$ on binary words.

[A1]

The statement refuted is: every effective binary encoding is prefix-free.

[L1]

An effective binary encoding is an injective map into ${0, 1}^{*}$ with a total decoder, and prefix-free means that no codeword is a proper prefix of another, by Effective binary encodings and total decoders.

Counterexample

technique · direct

1.1

The map $e (w) = 1 w$ is effective: define a total decoder by removing the first bit from any binary word beginning with $1$ , and sending every binary word not beginning with $1$ (including the empty word) to the fixed malformed token. Then the decoder recovers $w$ from $e (w) = 1 w$ .

givenL1

1.2

The codeword $1 = e (ε)$ is a proper prefix of $10 = e (0)$ , so $e$ is not prefix-free.

given

2.1

Therefore $e$ is a counterexample to [A1].

A1step 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

Michael Sipser, MIT 18.404J Theory of Computation, Lecture 7 (standard reference, not scraped)

The encoding w↦1ww↦1w is effective but not prefix-free