ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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 two-tree level product and dense matrix

Example

Let $T_{1} = T_{2} = 2^{< ω}$ , ordered by extension. The level product, the full product, and a dense matrix can be seen explicitly and are not the same notion.

Facts & Assumptions

Given: The two full binary trees in the example.

[F1]

The local definition distinguishes common-level products, full products, and coordinatewise $(h, k)$ -matrices. Finitistic trees, level products, density, and matrices

Verification

1.1

Since $T_{i} (2) = {00, 01, 10, 11}$ , its level-2 product consists of the sixteen pairs ${(00, 00), (00, 01), (00, 10), (00, 11), (01, 00), (01, 01), (01, 10), (01, 11), (10, 00), (10, 01), (10, 10), (10, 11), (11, 00), (11, 01), (11, 10), (11, 11)}$ . Every pair has common coordinate height $2$ .

F1given

1.2

For $h = 1, k = 2$ , use roots $0 \in T_{1} (1)$ and $1 \in T_{2} (1)$ and put $A_{1} = {000, 0010, 010, 011}$ and $A_{2} = {100, 101, 110, 111}$ . The height-3 frontier above $0$ is ${000, 001, 010, 011}$ ; the listed members of $A_{1}$ respectively dominate those four nodes. The height-3 frontier above $1$ is exactly $A_{2}$ . Thus both factors are $(1, 2)$ -dense, and $A_{1} \times A_{2}$ is a $(1, 2)$ -matrix.

F1construct

2.1

The pair $(0, 101)$ belongs to the full product $T_{1} \times T_{2}$ , but its coordinate heights are $1$ and $3$ , so it belongs to no common-level product.

F1step 1.1

3.1

This matrix is a subset of the full product but not of the level product: it contains $(0010, 100)$ , whose heights are $4$ and $3$ . Hence the level product imposes equal heights, the full product imposes none, and being a matrix imposes coordinatewise domination rather than equal height.

F1step 1.2step 2.1∎

Depends on

Finitistic trees, level products, density, and matrices

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Halpern–Läuchli, A partition theorem (1966), §1 definitions, pp. 360–361; explicit binary-tree instance (standard reference, not scraped)