Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck 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\mathbb N as in Sequences of reals: bounded, eventually, frequently, tails, subsequences, and comparisons use Order on the reals.

Facts & Assumptions

Given: A real sequence (xn)(x_n).

[L1]

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

[F1]

A sequence of reals is a function x:N→Rx : \mathbb{N} \to \mathbb{R} (Sequences of reals: bounded, eventually, frequently, tails, subsequences).

Verification

technique · direct
1.1

For i<ji<j, colour {i,j}\{i,j\} up when xi≤xjx_i\le x_j and down when xi>xjx_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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 45 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources