import Mathlib

namespace OAI

noncomputable section
open MeasureTheory Filter Set
open scoped ENNReal Topology BigOperators
open scoped BigOperators ComplexConjugate
open MeasureTheory ProbabilityTheory Filter Set
open scoped ENNReal NNReal Topology BigOperators
open MeasureTheory ProbabilityTheory Real Complex
open scoped ENNReal ComplexConjugate
open ProbabilityTheory
namespace PeriodicCutoff

lemma transition_derivative (x : ℝ) :
    HasDerivAt Real.smoothTransition (deriv Real.smoothTransition x) x :=
  ((Real.smoothTransition.contDiff (n := 1)).differentiable (by norm_num) x).hasDerivAt

lemma transition_deriv_eq_zero {x : ℝ} (hx : x∉Icc (0:ℝ) 1) :
    deriv Real.smoothTransition x=0 := by
  have he : (Real.smoothTransition =ᶠ[𝓝 x] fun _ ↦ (Real.smoothTransition x)) := by
    rcases not_and_or.mp hx with h | h
    · have hx : x<0 := lt_of_not_ge h
      filter_upwards [isOpen_Iio.mem_nhds hx] with y hy
      rw [Real.smoothTransition.zero_of_nonpos hy.le,Real.smoothTransition.zero_of_nonpos hx.le]
    · have hx : 1<x := lt_of_not_ge h
      filter_upwards [isOpen_Ioi.mem_nhds hx] with y hy
      rw [Real.smoothTransition.one_of_one_le hy.le,Real.smoothTransition.one_of_one_le hx.le]
  exact ((hasDerivAt_const x (Real.smoothTransition x)).congr_of_eventuallyEq he).deriv

lemma exists_transition_bound : ∃ M : ℝ, 0<M ∧ ∀ x : ℝ, |deriv Real.smoothTransition x|≤M := by
  obtain ⟨C,hC⟩ := isCompact_Icc.exists_bound_of_continuousOn
    ((Real.smoothTransition.contDiff (n := 2)).continuous_deriv (by norm_num)).continuousOn
      (s := Icc (0:ℝ) 1)
  refine ⟨max C 1,lt_of_lt_of_le zero_lt_one (le_max_right _ _),fun x ↦ ?_⟩
  by_cases hx : x∈Icc (0:ℝ) 1
  · exact (hC x hx).trans (le_max_left _ _)
  · rw [transition_deriv_eq_zero hx,abs_zero]
    exact (zero_le_one : (0:ℝ)≤1).trans (le_max_right _ _)

end PeriodicCutoff

end

end OAI