Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck 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.

x^x tends to one as x tends to zero from the right

Example

lim⁡x→0+xx=1.\lim_{x\to0+}x^x=1.

Facts & Assumptions

Given: x>0x>0 tending to 00.

[L1]

xx=exp⁡(xlog⁡x)x^x=\exp(x\log x) for x>0x>0 (Real powers for positive bases, with the zero-base positive-exponent convention).

[L2]

log⁡t/t→0\log t/t\to0 as t→+∞t\to+\infty (The logarithm grows more slowly than every positive real power).

[L3]

The exponential is continuous and exp⁡0=1\exp0=1 (The exponential function is strictly increasing).

Verification

technique · direct
1.1

With t=1/xt=1/x, one has t→+∞t\to+\infty and xlog⁡x=−(log⁡t)/t→0x\log x=-(\log t)/t\to0 by [L2].

L2
2.1

Therefore xx=exp⁡(xlog⁡x)→exp⁡0=1x^x=\exp(x\log x)\to\exp0=1.

step 1.1L1L3∎

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: 56 results over 17 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