ExampleConstruction: AI-adaptedVerification: 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.

Two to the square root of two from rational suprema and from exp(sqrt(2) log 2)

Example

Let $s = \sqrt{2}$ . The two constructions give the same number: $2^{[s]} = \sup {2^{q} : q \in Q, q < s} = \exp (s \log 2) = \lim_{q_{n} \to s, q_{n} \in Q} 2^{q_{n}} .$

Facts & Assumptions

Given: The nonnegative square root $s = \sqrt{2}$ .

[L1]

The rational-supremum and exponential constructions agree (The rational-supremum construction of real powers agrees with the exponential construction).

[L2]

Rational approximations to an exponent converge to its real power (Every rational approximation to a real exponent gives the same limiting real power).

[L3]

$\sqrt{2}$ exists and is positive (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ).

Verification

technique · direct

1.1

By [L3], $s$ is a real exponent, so [L1] gives $2^{[s]} = \exp (s \log 2)$ .

L1L3

1.2

For every rational sequence $q_{n} \to s$ , [L2] gives $2^{q_{n}} \to 2^{s} = \exp (s \log 2)$ .

2.1

Thus the supremum, exponential, and rational-limit descriptions agree.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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

J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)
MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)