-- Module: Mathlib.RingTheory.Kaehler.Polynomial
Goal:
R : Type u
inst : CommRing R
σ : Type u_1
a b : MvPolynomial σ R
single :
  ∀ (a : R),
    Eq (α := Ω[MvPolynomial σ R⁄R])
      ((Finsupp.linearCombination (MvPolynomial σ R) fun (x : σ) ↦
              (D R (MvPolynomial σ R) : (a : MvPolynomial σ R) → Ω[MvPolynomial σ R⁄R]) (MvPolynomial.X x) :
            (a : σ →₀ MvPolynomial σ R) → Ω[MvPolynomial σ R⁄R])
          ((MvPolynomial.mkDerivation R fun (x : σ) ↦ Finsupp.single x 1 :
              (a : MvPolynomial σ R) → σ →₀ MvPolynomial σ R)
            ((MvPolynomial.C : (a : R) → MvPolynomial σ R) a)) :
        Ω[MvPolynomial σ R⁄R])
      ((D R (MvPolynomial σ R) : (a : MvPolynomial σ R) → Ω[MvPolynomial σ R⁄R])
          ((MvPolynomial.C : (a : R) → MvPolynomial σ R) a) :
        Ω[MvPolynomial σ R⁄R])
single₁ :
  ∀ (p q : MvPolynomial σ R),
    Eq (α := Ω[MvPolynomial σ R⁄R])
        ((Finsupp.linearCombination (MvPolynomial σ R) fun (x : σ) ↦
                (D R (MvPolynomial σ R) : (a : MvPolynomial σ R) → Ω[MvPolynomial σ R⁄R]) (MvPolynomial.X x) :
              (a : σ →₀ MvPolynomial σ R) → Ω[MvPolynomial σ R⁄R])
            ((MvPolynomial.mkDerivation R fun (x : σ) ↦ Finsupp.single x 1 :
                (a : MvPolynomial σ R) → σ →₀ MvPolynomial σ R)
              p) :
          Ω[MvPolynomial σ R⁄R])
        ((D R (MvPolynomial σ R) : (a : MvPolynomial σ R) → Ω[MvPolynomial σ R⁄R]) p : Ω[MvPolynomial σ R⁄R]) →
      Eq (α := Ω[MvPolynomial σ R⁄R])
          ((Finsupp.linearCombination (MvPolynomial σ R) fun (x : σ) ↦
                  (D R (MvPolynomial σ R) : (a : MvPolynomial σ R) → Ω[MvPolynomial σ R⁄R]) (MvPolynomial.X x) :
                (a : σ →₀ MvPolynomial σ R) → Ω[MvPolynomial σ R⁄R])
              ((MvPolynomial.mkDerivation R fun (x : σ) ↦ Finsupp.single x 1 :
                  (a : MvPolynomial σ R) → σ →₀ MvPolynomial σ R)
                q) :
            Ω[MvPolynomial σ R⁄R])
          ((D R (MvPolynomial σ R) : (a : MvPolynomial σ R) → Ω[MvPolynomial σ R⁄R]) q : Ω[MvPolynomial σ R⁄R]) →
        Eq (α := Ω[MvPolynomial σ R⁄R])
          ((Finsupp.linearCombination (MvPolynomial σ R) fun (x : σ) ↦
                  (D R (MvPolynomial σ R) : (a : MvPolynomial σ R) → Ω[MvPolynomial σ R⁄R]) (MvPolynomial.X x) :
                (a : σ →₀ MvPolynomial σ R) → Ω[MvPolynomial σ R⁄R])
              ((MvPolynomial.mkDerivation R fun (x : σ) ↦ Finsupp.single x 1 :
                  (a : MvPolynomial σ R) → σ →₀ MvPolynomial σ R)
                (p + q)) :
            Ω[MvPolynomial σ R⁄R])
          ((D R (MvPolynomial σ R) : (a : MvPolynomial σ R) → Ω[MvPolynomial σ R⁄R]) (p + q) : Ω[MvPolynomial σ R⁄R])
single₂ :
  ∀ (p : MvPolynomial σ R) (n : σ),
    Eq (α := Ω[MvPolynomial σ R⁄R])
        ((Finsupp.linearCombination (MvPolynomial σ R) fun (x : σ) ↦
                (D R (MvPolynomial σ R) : (a : MvPolynomial σ R) → Ω[MvPolynomial σ R⁄R]) (MvPolynomial.X x) :
              (a : σ →₀ MvPolynomial σ R) → Ω[MvPolynomial σ R⁄R])
            ((MvPolynomial.mkDerivation R fun (x : σ) ↦ Finsupp.single x 1 :
                (a : MvPolynomial σ R) → σ →₀ MvPolynomial σ R)
              p) :
          Ω[MvPolynomial σ R⁄R])
        ((D R (MvPolynomial σ R) : (a : MvPolynomial σ R) → Ω[MvPolynomial σ R⁄R]) p : Ω[MvPolynomial σ R⁄R]) →
      Eq (α := Ω[MvPolynomial σ R⁄R])
        ((Finsupp.linearCombination (MvPolynomial σ R) fun (x : σ) ↦
                (D R (MvPolynomial σ R) : (a : MvPolynomial σ R) → Ω[MvPolynomial σ R⁄R]) (MvPolynomial.X x) :
              (a : σ →₀ MvPolynomial σ R) → Ω[MvPolynomial σ R⁄R])
            ((MvPolynomial.mkDerivation R fun (x : σ) ↦ Finsupp.single x 1 :
                (a : MvPolynomial σ R) → σ →₀ MvPolynomial σ R)
              (p * MvPolynomial.X n)) :
          Ω[MvPolynomial σ R⁄R])
        ((D R (MvPolynomial σ R) : (a : MvPolynomial σ R) → Ω[MvPolynomial σ R⁄R]) (p * MvPolynomial.X n) :
          Ω[MvPolynomial σ R⁄R])
⊢ Eq (α := Ω[MvPolynomial σ R⁄R])
    ((Finsupp.linearCombination (MvPolynomial σ R) fun (x : σ) ↦
            (D R (MvPolynomial σ R) : (a : MvPolynomial σ R) → Ω[MvPolynomial σ R⁄R]) (MvPolynomial.X x) :
          (a : σ →₀ MvPolynomial σ R) → Ω[MvPolynomial σ R⁄R])
        ((MvPolynomial.mkDerivation R fun (x : σ) ↦ Finsupp.single x 1 : (a : MvPolynomial σ R) → σ →₀ MvPolynomial σ R)
          a) :
      Ω[MvPolynomial σ R⁄R])
    ((D R (MvPolynomial σ R) : (a : MvPolynomial σ R) → Ω[MvPolynomial σ R⁄R]) a : Ω[MvPolynomial σ R⁄R])

Tactic:
induction a using MvPolynomial.induction_on

Premises used: MvPolynomial.induction_on