feat(RingTheory/MvPowerSeries): partial derivatives of MvPowerSeries (#39626)

Previously, we had formal derivatives for `PowerSeries` and `Polynomial` and formal partial derivatives for `MvPolynomial`, but no formal partial derivatives for `MvPowerSeries`. This PR adds them. Furthermore, `PowerSeries.derivative` is refactored to be defined in terms of `MvPowerSeries.pderiv`, which reduces code duplication (In particular, there is no need to define the bare function `PowerSeries.derivativeFun` anymore).

Most proofs are direct generalizations from the univariate case. I only had to add a few missing API lemmas, which I am PR'ing separately below:

- [x] depends on: #39623 
- [x] depends on: #39624 
- [x] depends on: #39625

Author: justus-springer

Mathlib.lean

@@ -6830,6 +6830,7 @@ public import Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities
 public import Mathlib.RingTheory.MvPolynomial.Tower
 public import Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous
 public import Mathlib.RingTheory.MvPowerSeries.Basic
+public import Mathlib.RingTheory.MvPowerSeries.Derivative
 public import Mathlib.RingTheory.MvPowerSeries.Equiv
 public import Mathlib.RingTheory.MvPowerSeries.Evaluation
 public import Mathlib.RingTheory.MvPowerSeries.Expand

Mathlib/RingTheory/MvPowerSeries/Derivative.lean

@@ -0,0 +1,209 @@
+/-
+Copyright (c) 2026 Justus Springer. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Justus Springer
+-/
+module
+
+public import Mathlib.Algebra.MvPolynomial.PDeriv
+public import Mathlib.RingTheory.MvPowerSeries.Inverse
+public import Mathlib.RingTheory.MvPowerSeries.Trunc
+
+/-!
+# Formal partial derivatives of multivariate power series
+
+This file defines `MvPowerSeries.pderiv R i`, the formal partial derivative of a multivariate
+power series with respect to variable `i`, as a
+`Derivation R (MvPowerSeries σ R) (MvPowerSeries σ R)`.
+
+See also `PowerSeries.derivative` for the univariate setting.
+
+## Main definitions
+
+- `MvPowerSeries.pderiv R i`: the formal partial derivative with respect to `i`, as a derivation.
+
+## Main results
+
+- `MvPowerSeries.coeff_pderiv`: coefficient formula
+  `coeff n (pderiv R i f) = coeff (n + single i 1) f * (n i + 1)`.
+- `MvPowerSeries.pderiv_coe`: compatibility with `MvPolynomial.pderiv`.
+- `MvPowerSeries.trunc_pderiv`: truncation commutes with partial differentiation.
+- `MvPowerSeries.pderiv.ext`: a power series is determined by its constant term and its partial
+  derivatives.
+- `MvPowerSeries.pderiv_pow`: power rule.
+- `MvPowerSeries.pderiv_inv`, `MvPowerSeries.pderiv_inv'`: derivative of an inverse.
+
+-/
+
+@[expose] public section
+
+namespace MvPowerSeries
+
+open MvPolynomial Finsupp
+
+variable {σ R : Type*}
+
+section Semiring
+
+variable [Semiring R]
+
+/-- The underlying function of the formal partial derivative with respect to variable `i`.
+This is packaged as a derivation in `MvPowerSeries.pderiv`. -/
+noncomputable def pderivFun (i : σ) (f : MvPowerSeries σ R) : MvPowerSeries σ R :=
+  fun d ↦ coeff (d + single i 1) f * (d i + 1)
+
+theorem coeff_pderivFun {i : σ} (f : MvPowerSeries σ R) (d : σ →₀ ℕ) :
+    coeff d (f.pderivFun i) = coeff (d + single i 1) f * (d i + 1) := by
+  rfl
+
+theorem pderivFun_add {i : σ} (f g : MvPowerSeries σ R) :
+    pderivFun i (f + g) = pderivFun i f + pderivFun i g := by
+  ext
+  rw [coeff_pderivFun, map_add, map_add, coeff_pderivFun, coeff_pderivFun, add_mul]
+
+theorem pderivFun_C {i : σ} (r : R) : pderivFun i (C r) = 0 := by
+  ext n
+  rw [coeff_pderivFun, coeff_add_single_C, zero_mul, (coeff n).map_zero]
+
+theorem pderivFun_one {i : σ} : pderivFun i (1 : MvPowerSeries σ R) = 0 := by
+  rw [← map_one C, pderivFun_C (1 : R)]
+
+end Semiring
+
+section CommSemiring
+
+variable [CommSemiring R]
+
+private theorem pderivFun_coe {i : σ} (f : MvPolynomial σ R) :
+    (f : MvPowerSeries σ R).pderivFun i = f.pderiv i := by
+  ext
+  rw [coeff_pderivFun, coeff_coe, coeff_coe, coeff_pderiv]
+
+private theorem trunc_pderivFun [DecidableEq σ] {i : σ} (f : MvPowerSeries σ R) (n : σ →₀ ℕ) :
+    trunc R n (pderivFun i f) = pderiv i (trunc R (n + single i 1) f) := by
+  ext
+  rw [coeff_trunc]
+  split_ifs with h
+  · rw [coeff_pderivFun, coeff_pderiv, coeff_trunc, if_pos (add_lt_add_left h _)]
+  · rw [coeff_pderiv, coeff_trunc, if_neg ((add_lt_add_iff_right _).not.mpr h), zero_mul]
+
+-- A special case of `pderivFun_mul`, used in its proof.
+private theorem pderivFun_coe_mul_coe {i : σ} (f g : MvPolynomial σ R) :
+    pderivFun i (f * g : MvPowerSeries σ R) = f * pderiv i g + g * pderiv i f := by
+  rw [← coe_mul, pderivFun_coe, pderiv_mul, add_comm, mul_comm _ g, ← coe_mul, ← coe_mul,
+    MvPolynomial.coe_add]
+
+private theorem pderivFun_mul {i : σ} (f g : MvPowerSeries σ R) :
+    pderivFun i (f * g) = f • g.pderivFun i + g • f.pderivFun i := by
+  classical
+  ext n
+  have h₁ : n < n + single i 1 := lt_def.mpr ⟨self_le_add_right _ _, i, by simp⟩
+  have h₂ : n + single i 1 < n + single i 1 + single i 1 :=
+    lt_def.mpr ⟨self_le_add_right _ _, i, by simp⟩
+  have h₃ : n < n + single i 1 + single i 1 := lt_trans h₁ h₂
+  rw [coeff_pderivFun, map_add, ← coeff_trunc_mul_trunc_eq_coeff_mul _ _ _ h₂, smul_eq_mul,
+    smul_eq_mul, ← coeff_trunc_mul_trunc_eq_coeff_mul₂ _ _ g (f.pderivFun i) h₃ h₁,
+    ← coeff_trunc_mul_trunc_eq_coeff_mul₂ _ _ f (g.pderivFun i) h₃ h₁, trunc_pderivFun,
+    trunc_pderivFun, ← coeff_coe, ← coeff_coe, ← coeff_coe, ← map_add, coe_mul, coe_mul, coe_mul,
+    ← pderivFun_coe_mul_coe, coeff_pderivFun]
+
+private theorem pderivFun_smul {i : σ} (r : R) (f : MvPowerSeries σ R) :
+    pderivFun i (r • f) = r • pderivFun i f := by
+  rw [smul_eq_C_mul, smul_eq_C_mul, pderivFun_mul, pderivFun_C, smul_zero, add_zero, smul_eq_mul]
+
+variable (R) in
+/-- The formal partial derivative of a multivariate formal power series with respect to
+variable `i`, as an `R`-derivation on `MvPowerSeries σ R`. -/
+@[no_expose]
+noncomputable def pderiv (i : σ) : Derivation R (MvPowerSeries σ R) (MvPowerSeries σ R) where
+  toFun := pderivFun i
+  map_add' := pderivFun_add
+  map_smul' := pderivFun_smul
+  map_one_eq_zero' := pderivFun_one
+  leibniz' := pderivFun_mul
+
+@[simp] theorem pderiv_C {i : σ} {r : R} : pderiv R i (C r) = 0 := pderivFun_C r
+
+theorem pderiv_one {i : σ} : pderiv R i 1 = 0 := pderiv_C
+
+theorem coeff_pderiv {i : σ} (f : MvPowerSeries σ R) (n : σ →₀ ℕ) :
+    coeff n (pderiv R i f) = coeff (n + single i 1) f * (n i + 1) :=
+  coeff_pderivFun f n
+
+theorem pderiv_coe {i : σ} (f : MvPolynomial σ R) :
+    pderiv R i f = MvPolynomial.pderiv i f := pderivFun_coe f
+
+@[simp]
+theorem pderiv_X_self {i : σ} : pderiv R i (X i) = 1 := by
+  classical
+  ext n
+  simp only [coeff_pderiv, coeff_X, boole_mul, add_eq_right, coeff_one]
+  split_ifs <;> simp_all
+
+@[simp]
+theorem pderiv_X_of_ne {i j : σ} (h : j ≠ i) : pderiv R i (X j) = 0 := by
+  classical
+  ext n
+  simpa only [coeff_pderiv, coeff_X, boole_mul, coeff_zero] using
+    if_neg (ne_iff.mpr ⟨i, by grind [Finsupp.add_apply]⟩)
+
+theorem pderiv_X [DecidableEq σ] (i j : σ) :
+    pderiv R i (X j) = Pi.single (M := fun _ => MvPowerSeries σ R) i 1 j := by
+  by_cases h : i = j
+  · subst h; simp only [pderiv_X_self, Pi.single_eq_same]
+  · grind [pderiv_X_of_ne]
+
+theorem trunc_pderiv [DecidableEq σ] {i : σ} (f : MvPowerSeries σ R) (n : σ →₀ ℕ) :
+    trunc R n (pderiv R i f) = MvPolynomial.pderiv i (trunc R (n + single i 1) f) :=
+  trunc_pderivFun ..
+
+/-- The partial derivative of `g^n` equals `n * g^(n-1) * g'`. -/
+theorem pderiv_pow {i : σ} (g : MvPowerSeries σ R) (n : ℕ) :
+    pderiv R i (g ^ n) = n * g ^ (n - 1) * pderiv R i g := by
+  rw [Derivation.leibniz_pow, smul_eq_mul, nsmul_eq_mul, mul_assoc]
+
+end CommSemiring
+
+/-- If `f` and `g` have the same constant term and all partial derivatives, then they are equal.
+
+The `CommRing` assumption is needed because the proof uses `smul_right_inj`, which requires
+cancellation of addition in `R`; `IsAddTorsionFree` alone does not suffice. -/
+theorem pderiv.ext [CommRing R] [IsAddTorsionFree R] {f g : MvPowerSeries σ R}
+    (hD : ∀ i, pderiv R i f = pderiv R i g) (hc : constantCoeff f = constantCoeff g) : f = g := by
+  ext n
+  by_cases h : n = 0
+  · rw [h, coeff_zero_eq_constantCoeff, hc]
+  obtain ⟨i, hi : n i ≠ 0⟩ := ne_iff.mp h
+  have : single i 1 ≤ n := fun j ↦ by
+    by_cases hj : j = i <;> grind [single_eq_same, single_eq_of_ne]
+  have e := congr(coeff (n - single i 1) $(hD i))
+  rwa [coeff_pderiv, coeff_pderiv, tsub_add_cancel_of_le this, coe_tsub, Pi.sub_apply,
+    single_eq_same, Nat.cast_sub (Nat.one_le_iff_ne_zero.mpr hi), Nat.cast_one, sub_add_cancel,
+    mul_comm, ← nsmul_eq_mul, mul_comm, ← nsmul_eq_mul, smul_right_inj hi] at e
+
+@[simp]
+theorem pderiv_inv {i : σ} [CommRing R] (f : (MvPowerSeries σ R)ˣ) :
+    pderiv R i ↑f⁻¹ = -(↑f⁻¹ : MvPowerSeries σ R) ^ 2 * pderiv R i f :=
+  (pderiv R i).leibniz_of_mul_eq_one f.inv_mul
+
+@[simp]
+theorem pderiv_invOf {i : σ} [CommRing R] (f : MvPowerSeries σ R) [Invertible f] :
+    pderiv R i ⅟f = -⅟f ^ 2 * pderiv R i f :=
+  (pderiv R i).leibniz_invOf f
+
+/-
+The following theorem is stated only in the case that `R` is a field. This is because
+there is currently no instance of `Inv (MvPowerSeries σ R)` for more general base rings `R`.
+-/
+
+@[simp]
+theorem pderiv_inv' {i : σ} [Field R] (f : MvPowerSeries σ R) :
+    pderiv R i f⁻¹ = -f⁻¹ ^ 2 * pderiv R i f := by
+  by_cases h : constantCoeff f = 0
+  · suffices f⁻¹ = 0 by
+      rw [this, pow_two, zero_mul, neg_zero, zero_mul, map_zero]
+    rwa [MvPowerSeries.inv_eq_zero]
+  apply Derivation.leibniz_of_mul_eq_one
+  exact MvPowerSeries.inv_mul_cancel (h := h)
+
+end MvPowerSeries