nevanlinna/Nevanlinna/logpos.lean

import Mathlib.Analysis.SpecialFunctions.Log.Basic

open Real


noncomputable def logpos : ℝ → ℝ := fun r ↦ max 0 (log r)

notation "log⁺" => logpos


theorem loglogpos {r : ℝ} : log r = log⁺ r - log⁺ r⁻¹ := by
  unfold logpos
  rw [log_inv]
  by_cases h : 0 ≤ log r
  · simp [h]
  · simp at h
    have : 0 ≤ -log r := Left.nonneg_neg_iff.2 (le_of_lt h)
    simp [h, this]
    exact neg_nonneg.mp this


theorem logpos_norm {r : ℝ} : log⁺ r = 2⁻¹ * (log r + ‖log r‖) := by
  by_cases hr : 0 ≤ log r
  · rw [norm_of_nonneg hr]
    have : logpos r = log r := by
      unfold logpos
      simp [hr]
    rw [this]
    ring
  · rw [norm_of_nonpos (le_of_not_ge hr)]
    have : logpos r = 0 := by
      unfold logpos
      simp
      exact le_of_not_ge hr
    rw [this]
    ring

-- WANT: logpos is even and continuous
-- WANT: Inequalities