Compatible derivations obey the intermediate-field product rule

Proved

OAI.PiExponent.derivation_intermediate_smul

by marwahaha · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

differential-algebranumber-theoryopenai-pi-exponent-017

Let $k\subseteq F\subseteq E$ be a compatible tower of fields, with compatible $k$ -derivations $d:F\to F$ and $D:E\to E$ . For every $a\in F$ and $z\in E$ ,

D(a\cdot z)=d(a)\cdot z+a\cdot D(z).

Here multiplication by an element of $F$ uses its given action on $E$ .

Preamble

import Mathlib.Algebra.Polynomial.Derivation
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory.LaurentSeries
import Mathlib.RingTheory.Trace.Defs
import Mathlib.Tactic


namespace OAI
namespace PiExponent
end PiExponent
end OAI
open scoped LaurentSeries Matrix
open Polynomial
open OAI.PiExponent

Formal statement

theorem OAI.PiExponent.derivation_intermediate_smul
    {k F E : Type*} [Field k] [Field F] [Field E]
    [Algebra k F] [Algebra k E] [Algebra F E] [IsScalarTower k F E]
    (d : Derivation k F F) (D : Derivation k E E)
    (hcompat : ∀ a : F, D (algebraMap F E a) = algebraMap F E (d a))
    (a : F) (z : E) : D (a • z) = d a • z + a • D z := by sorry

Source

https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/NumberTheory/PiExponent/Analysis/LogarithmicObstruction.lean#L101-L109

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