LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Basic conditional-complexity inequalities

Statement

For finite strings $x, y$ , $C (x ∣ y) \leq C (x) + O (1)$ and $C (x, y) \leq C (x) + C (y ∣ x) + O (\log (C (x) + 2) + \log (C (y ∣ x) + 2))$ .

Facts & Assumptions

Given: a fixed optimal plain machine $U$ , the fixed optimal conditional machine $V$ , a fixed effective encoding $⟨ x, y ⟩$ of pairs, and shortest relevant descriptions. Write $C (x, y) : = C_{U} (⟨ x, y ⟩)$ .

Proof

1.1

The conditional machine $D (p, y) = U (p)$ ignores its condition. Conditional optimality therefore gives $C (x ∣ y) \leq C (x) + O (1)$ .

givenconstruct

2.1

Let $p$ be a shortest $U$ -description of $x$ and $q$ a shortest $V$ -description of $y$ conditional on $x$ . Self-delimit $∣ p ∣$ and $∣ q ∣$ and concatenate the two programs. A fixed plain description machine recovers $x = U (p)$ , then $y = V (q, x)$ , and outputs $⟨ x, y ⟩$ . Its description length is $∣ p ∣ + ∣ q ∣ + O (\log (∣ p ∣ + 2) + \log (∣ q ∣ + 2)) .$ Optimality of $U$ , supplied by Invariance theorem for plain Kolmogorov complexity, transfers this bound to $C (x, y)$ and proves the second inequality.

givenconstruct∎

Depends on

Used by

Symmetry of information up to logarithmic terms Theorem

Dependency tree · two levels

7 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

Shen, §13 (standard reference, not scraped)