Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generated
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.

LZ78 parsing of a finite binary word

Example

Traces the specified parser and decoder on 0100101001 followed by the explicit terminal symbol, including dictionary indices and exact record sequence.

Facts & Assumptions

Given: The LZ78 parser of Finite-block LZ78 incremental prefix code, input 0100101001, and terminal symbol #.

[F1]

Finite-block LZ78 incremental prefix code: output (i,c) appends dictionary word D[i]c; the terminator stops parsing and is not added.

[F2]

Losslessness and prefix property of the explicit LZ78 code: the explicit records decode uniquely, with the terminator ending the codeword.

Verification

Given: The input string, empty root dictionary D[0]=ϵ, and conventions [F1]–[F2].

1.1F1givenalgebra

Applying the longest-dictionary-prefix rule successively gives phrases 0,1,00,10,100,1#. The records are respectively (0,0),(0,1),(1,0),(2,0),(4,0),(2,#); after the first five records the dictionary indices 1 through 5 name 0,1,00,10,100. The sixth record terminates without insertion.

2.1F1step 1.1construct

The dictionary trie has root index 0, edges 0 and 1 to nodes 1 and 2, then edge 0 from node 1 to node 3, edge 0 from node 2 to node 4, and edge 0 from node 4 to node 5. The dashed edge marked # from node 2 is the final output phrase only; it is not a dictionary node. Each node label gives its dictionary index and phrase; each edge label is the symbol appended to the parent's phrase, and the root phrase is empty. The diagram's ordinary nodes are exactly the phrases inserted after the first five records; the terminal edge is drawn dashed because [F1] does not insert it.

3.1F1F2step 1.1step 2.1algebra∎

Starting with the root, the decoder expands the listed records to the same six phrases; concatenating them yields 0100101001#. It stops at #, so deleting the marker recovers the input exactly, as guaranteed by [F2].

Remarks

Diagram

The trie uses the dictionary indices and appended-symbol edges described in the verification; the dashed terminal edge is output only.

diagram failed to render:
\begin{tikzpicture}[every node/.style={draw,circle,inner sep=2pt},>=stealth]
\node (r) at (0,0) {$0:\epsilon$};
\node (a) at (-2,-1) {$1:0$};
\node (b) at (2,-1) {$2:1$};
\node (aa) at (-2,-2) {$3:00$};
\node (ba) at (2,-2) {$4:10$};
\node (baa) at (2,-3) {$5:100$};
\node[draw=none,rectangle] (end) at (4,-2) {$1\#: \text{output only}$};
\draw (r)--node[above left,draw=none] {$0$}(a);
\draw (r)--node[above right,draw=none] {$1$}(b);
\draw (a)--node[left,draw=none] {$0$}(aa);
\draw (b)--node[right,draw=none] {$0$}(ba);
\draw (ba)--node[right,draw=none] {$0$}(baa);
\draw[dashed,->] (b)--node[above,draw=none] {$\#$}(end);
\end{tikzpicture}

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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