TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-02

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.

Young's inequality for conjugate real exponents

Let $p, q > 1$ satisfy $1 / p + 1 / q = 1$ . For $u, v \geq 0$ , $u v \leq \frac{u^{p}}{p} + \frac{v^{q}}{q} .$

Given: Conjugate real exponents $p, q > 1$ and nonnegative reals $u, v$ .

[L1]

Weighted AM-GM applies to positive entries with nonnegative weights summing to one (The weighted arithmetic-geometric mean inequality for real weights).

[L2]

technique · direct

1.1L2

If $u = 0$ or $v = 0$ , the inequality is immediate from nonnegativity of the two terms on the right.

1.2L1given

For $u, v > 0$ , apply [L1] to $u^{p}, v^{q}$ with weights $1 / p, 1 / q$ .

2.1step 1.2L2

Its geometric side is $(u^{p})^{1 / p} (v^{q})^{1 / q} = u v$ , and its arithmetic side is $u^{p} / p + v^{q} / q$ .

3.1step 1.1step 2.1∎

Steps 1.1 and 2.1 cover all nonnegative $u, v$ .

Depends on