import Mathlib.Data.Fin.Tuple.Basic
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Complex.TaylorSeries
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Symmetric


noncomputable def Real.partialDeriv : ℂ → (ℂ → ℂ) → (ℂ → ℂ) := by
  intro v
  intro f
  exact fun w ↦ (fderiv ℝ f w) v


theorem partialDeriv_smul {f : ℂ → ℂ } {a v : ℂ} (h : Differentiable ℝ f) : Real.partialDeriv v (a • f) = a • Real.partialDeriv v f := by
  unfold Real.partialDeriv

  have : a • f = fun y ↦ a • f y := by rfl
  rw [this]

  conv =>
    left
    intro w
    rw [fderiv_const_smul (h w)]


theorem partialDeriv_contDiff {n : ℕ} {f : ℂ → ℂ} (h : ContDiff ℝ (n + 1) f) : ∀ v : ℂ, ContDiff ℝ n (Real.partialDeriv v f) := by
  unfold Real.partialDeriv
  intro v

  let A := (contDiff_succ_iff_fderiv.1 h).right
  simp at A

  have : (fun w => (fderiv ℝ f w) v) = (fun f => f v) ∘ (fun w => (fderiv ℝ f w)) := by
    rfl

  rw [this]
  refine ContDiff.comp ?hg A
  refine ContDiff.of_succ ?hg.h
  refine ContDiff.clm_apply ?hg.h.hf ?hg.h.hg
  exact contDiff_id
  exact contDiff_const