nevanlinna/Nevanlinna/mathlibAddOn.lean

import Mathlib.Analysis.Analytic.Meromorphic
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.FDeriv.Add
import Nevanlinna.analyticAt

variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
variable {G : Type*} [NormedAddCommGroup G] [NormedSpace 𝕜 G]
variable {G' : Type*} [NormedAddCommGroup G'] [NormedSpace 𝕜 G']
variable {f f₀ f₁ g : E → F}
variable {f' f₀' f₁' g' : E →L[𝕜] F}
variable (e : E →L[𝕜] F)
variable {x : E}
variable {s t : Set E}
variable {L L₁ L₂ : Filter E}

variable {R : Type*} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F]


-- import Mathlib.Analysis.Calculus.FDeriv.Add

@[fun_prop]
theorem Differentiable.const_smul' (h : Differentiable 𝕜 f) (c : R) :
    Differentiable 𝕜 (c • f) := by
  have : c • f = fun x ↦ c • f x := rfl
  rw [this]
  exact Differentiable.const_smul h c


-- Mathlib.Analysis.Calculus.ContDiff.Basic

theorem ContDiff.const_smul' {f : E → F} (c : R) (hf : ContDiff 𝕜 n f) :
    ContDiff 𝕜 n (c • f) := by
  have : c • f = fun x ↦ c • f x := rfl
  rw [this]
  exact ContDiff.const_smul c hf


-- Mathlib.Analysis.Analytic.Meromorphic

theorem meromorphicAt_congr
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜]
  {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
  {f : 𝕜 → E} {g : 𝕜 → E} {x : 𝕜}
  (h : f =ᶠ[nhdsWithin x {x}ᶜ] g) : MeromorphicAt f x ↔ MeromorphicAt g x :=
  ⟨fun hf ↦ hf.congr h, fun hg ↦ hg.congr h.symm⟩

theorem meromorphicAt_congr'
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜]
  {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
  {f : 𝕜 → E} {g : 𝕜 → E} {x : 𝕜}
  (h : f =ᶠ[nhds x] g) : MeromorphicAt f x ↔ MeromorphicAt g x :=
  meromorphicAt_congr (Filter.EventuallyEq.filter_mono h nhdsWithin_le_nhds)

open Topology Filter

lemma Mnhds
  {α : Type}
  {f g : ℂ → α}
  {z₀ : ℂ}
  (h₁ : f =ᶠ[𝓝[≠] z₀] g)
  (h₂ : f z₀ = g z₀) :
  f =ᶠ[𝓝 z₀] g := by
  apply eventually_nhds_iff.2
  obtain ⟨t, h₁t, h₂t⟩ := eventually_nhds_iff.1 (eventually_nhdsWithin_iff.1 h₁)
  use t
  constructor
  · intro y hy
    by_cases h₂y : y ∈ ({z₀}ᶜ : Set ℂ)
    · exact h₁t y hy h₂y
    · simp at h₂y
      rwa [h₂y]
  · exact h₂t