Exact source context

import Mathlib.Analysis.Calculus.Deriv.MeanValue
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Tactic

/-!
# Exponential-kernel interpolation

This module isolates the real-analysis step shared by semigroup entropy and
gradient estimates.  A differentiable interpolation whose derivative is
bounded by a backward exponential kernel admits the corresponding exact
endpoint convolution bound.
-/

namespace AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities

open Filter Set

noncomputable section

/-- Integrating a derivative bound with a backward exponential kernel gives
the exact endpoint convolution factor.

The quantities `source` and `energy` may have either sign: the theorem assumes
the required derivative inequality directly.  Algebraically, a nonzero rate
is enough for the proof; strict positivity records the intended decay regime. -/
theorem endpoint_le_of_deriv_le_exponential_kernel
    {H H' : ℝ → ℝ} {rate source t energy : ℝ}
    (hrate : 0 < rate) (ht : 0 ≤ t)
    (hH : ∀ s ∈ Icc (0 : ℝ) t, HasDerivAt H (H' s) s)
    (hzero : H 0 = 0)
    (hderiv : ∀ s ∈ Ioo (0 : ℝ) t,
      H' s ≤ source * Real.exp (-rate * (t - s)) * energy) :
    H t ≤ (source / rate) * (1 - Real.exp (-rate * t)) * energy := by
  let B : ℝ → ℝ := fun s =>
    (source / rate) *
      (Real.exp (-rate * (t - s)) - Real.exp (-rate * t)) * energy
  have hB : ∀ s : ℝ,
      HasDerivAt B
        (source * Real.exp (-rate * (t - s)) * energy) s := by
    intro s
    have hinner : HasDerivAt (fun u : ℝ => -rate * (t - u)) rate s := by
      have hcoeff : -(-rate) = rate := by ring
      simpa only [Pi.sub_apply, id_eq, zero_sub, mul_neg, mul_one, hcoeff] using
        (((hasDerivAt_const s t).sub (hasDerivAt_id s)).const_mul (-rate))
    have hexp := (Real.hasDerivAt_exp (-rate * (t - s))).comp s hinner
    have hsub := hexp.sub_const (Real.exp (-rate * t))
    have hmul := (hsub.const_mul (source / rate)).mul_const energy
    convert hmul using 1 <;> try rfl
    field_simp [ne_of_gt hrate]
  have hmono : MonotoneOn (fun s => B s - H s) (Icc (0 : ℝ) t) := by
    apply monotoneOn_of_deriv_nonneg (convex_Icc (0 : ℝ) t)
    · exact continuousOn_of_forall_continuousAt fun s hs =>
        ((hB s).sub (hH s hs)).continuousAt
    · intro s hs
      exact ((hB s).sub (hH s (interior_subset hs))).differentiableAt.differentiableWithinAt
    · intro s hs
      change 0 ≤ deriv (B - H) s
      rw [((hB s).sub (hH s (interior_subset hs))).deriv]
      exact sub_nonneg.mpr (hderiv s (by simpa only [interior_Icc] using hs))
  have hend := hmono ⟨le_rfl, ht⟩ ⟨ht, le_rfl⟩ ht
  dsimp [B] at hend
  have hBzero :
      (source / rate) *
        (Real.exp (-rate * (t - 0)) - Real.exp (-rate * t)) * energy = 0 := by
    ring_nf
  rw [hBzero, hzero] at hend
  have hBt :
      (source / rate) *
        (Real.exp (-rate * (t - t)) - Real.exp (-rate * t)) * energy =
      (source / rate) * (1 - Real.exp (-rate * t)) * energy := by
    rw [sub_self, mul_zero, Real.exp_zero]
  rw [hBt] at hend
  exact sub_nonneg.mp (by simpa only [sub_self] using hend)

end

end AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities