Plain-English statement
The pointwise Renyi integrand p(t)^a q(t)^(1-a) has the expected product-rule derivative when p and q are differentiable and nonzero.
Mathematical statement
d/dt [p^a q^(1-a)] is the sum of the differentiated p-power and q-power terms shown in the Lean statement.
Intuition
This is only the pointwise calculus leaf. Differentiating a Renyi integral also needs positivity, measurability, integrability, and a dominating function.
Conditions
- p and q have derivatives at t.
- p(t) and q(t) are nonzero for the real-power derivative API.
Why these conditions cannot be dropped
- Real powers have singular derivative behavior at zero for general exponents.
- A pointwise derivative does not justify differentiation under the integral.
Proof route
- Apply the constant real-power derivative rule to p.
- Apply it to q with exponent 1-a.
- Use the product derivative rule and normalize exponents.
Lean interface notes
- HasDerivAt.rpow_const exposes the nonzero side condition.
- The integral and ENNReal finiteness layers are separate declarations.
Read the mathematics first, then descend into Lean
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Lean statement
theorem hasDerivAt_renyiIntegrand {a : ℝ} {p q : ℝ → ℝ} {t pdot qdot : ℝ}
(hp : HasDerivAt p pdot t) (hq : HasDerivAt q qdot t)
(hp0 : p t ≠ 0) (hq0 : q t ≠ 0) :
HasDerivAt (fun s => renyiIntegrand a (p s) (q s))
((pdot * a * (p t) ^ (a - 1)) * (q t) ^ (1 - a) +
(p t) ^ a * (qdot * (1 - a) * (q t) ^ (-a))) t := by
have hpPow :
HasDerivAt (fun s => p s ^ a) (pdot * a * (p t) ^ (a - 1)) t :=
hp.rpow_const (p := a) (Or.inl hp0)
have hqPow :
HasDerivAt (fun s => q s ^ (1 - a))
(qdot * (1 - a) * (q t) ^ ((1 - a) - 1)) t :=
hq.rpow_const (p := 1 - a) (Or.inl hq0)
change HasDerivAt ((fun s => p s ^ a) * fun s => q s ^ (1 - a))
((pdot * a * (p t) ^ (a - 1)) * (q t) ^ (1 - a) +
(p t) ^ a * (qdot * (1 - a) * (q t) ^ (-a))) t
have hexponent : (1 - a) - 1 = -a := by ring
rw [hexponent] at hqPow
exact hpPow.mul hqPow
end Renyi
end InformationTheory
end TechnicalLemmas
end AutoSamplingTheoryOpen AutoSamplingTheory/TechnicalLemmas/InformationTheory/Renyi.lean:80published source at 7bcd37294df1