LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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.

The logarithm tangent inequality

Statement

For every $t > 0$ , $\ln t \leq t - 1$ , with equality exactly when $t = 1$ .

Facts & Assumptions

Given: $t > 0$ .

Proof

technique · direct

1.1givenalgebra

If $t \geq 1$ , then $1 / u \leq 1$ for $u \in [1, t]$ . Thus $\ln t = \int_{1}^{t} d u / u \leq t - 1$ , with strict inequality for $t > 1$ because $1 / u < 1$ on the nonempty interval $(1, t)$ .

2.1givenalgebra∎

If $0 < t < 1$ , then $1 / u > 1$ on $(t, 1)$ , so $\ln t = - \int_{t}^{1} d u / u < - (1 - t) = t - 1$ . At $t = 1$ both sides are zero. This proves the claim and equality case.

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Ronald de Wolf, Quantum Computing: Lecture Notes, arXiv:1907.09415v6 (standard reference, not scraped)