ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-11

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.

Infinite Ramsey for pairs gives a nondecreasing or nonincreasing subsequence of every real sequence

Example

Every real sequence has a nondecreasing or nonincreasing subsequence. Here a sequence is indexed by $N$ as in Sequences of reals: bounded, eventually, frequently, tails, subsequences, and comparisons use Order on the reals.

Facts & Assumptions

Given: A real sequence $(x_{n})$ .

[L1]

Every finite colouring of $[N]^{k}$ has an infinite monochromatic set, in ZF (Infinite Ramsey theorem on $N$ : every finite colouring of $[N]^{k}$ has an infinite monochromatic set, in ZF).

[F1]

A sequence of reals is a function $x : N \to R$ (Sequences of reals: bounded, eventually, frequently, tails, subsequences).

Verification

technique · direct

1.1

For $i < j$ , colour ${i, j}$ up when $x_{i} \leq x_{j}$ and down when $x_{i} > x_{j}$ . By [L1] there is an infinite homogeneous set of indices.

L1F1

2.1

Enumerate that set increasingly. In the up case every earlier selected term is at most every later one, giving a nondecreasing subsequence. In the down case every earlier selected term is greater than every later one, giving a strictly decreasing, hence nonincreasing, subsequence.

step 1.1F1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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

I. B. Leader, Ramsey Theory, example after Theorem 1 (standard reference, not scraped)