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.

The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents

For $a, b > 0$ and $r, s \in R$ , $a^{r + s} = a^{r} a^{s}, (a b)^{r} = a^{r} b^{r}, (a / b)^{r} = a^{r} / b^{r}, (a^{r})^{s} = a^{r s} .$

Given: Positive reals $a, b$ and real exponents $r, s$ .

[L1]

[L2]

[L3]

technique · direct

1.1L1L3

Expanding $a^{r + s}$ by [L1] and applying [L3] gives $a^{r + s} = \exp (r \log a) \exp (s \log a) = a^{r} a^{s}$ .

1.2L1L2L3

Expanding $(a b)^{r}$ and using $\log (a b) = \log a + \log b$ gives $(a b)^{r} = a^{r} b^{r}$ .

1.3L1L2L3

The same calculation with $\log (a / b) = \log a - \log b$ and [L3] gives $(a / b)^{r} = a^{r} / b^{r}$ .

2.1L1L2∎

Since $\log (a^{r}) = \log (\exp (r \log a)) = r \log a$ , expanding $(a^{r})^{s}$ gives $a^{r s}$ .

Depends on