exponential_decay_of_scaled_dissipation_from

Plain-English statement

- Coercivity plus exact dissipation gives exponential decay between any two times `s ≤ t`.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

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Lean statement

theorem exponential_decay_of_scaled_dissipation_from
    {scale rate : ℝ} (curve : DissipationCurve scale)
    (hcoercive : ∀ u : ℝ,
      rate * curve.energy u ≤ scale * curve.dissipation u)
    {s t : ℝ} (hst : s ≤ t) :
    curve.energy t ≤
      curve.energy s * Real.exp (-rate * (t - s)) := by
  have hbound : ∀ x ∈ Ico s t,
      -scale * curve.dissipation x ≤
        (-rate) * curve.energy x + 0 := by
    intro x hx
    have hneg := neg_le_neg (hcoercive x)
    simpa [neg_mul] using hneg
  have hgronwall :=
    le_gronwallBound_of_liminf_deriv_right_le
      (f := curve.energy)
      (f' := fun x => -scale * curve.dissipation x)
      (δ := curve.energy s)
      (K := -rate)
      (ε := 0)
      (a := s)
      (b := t)
      curve.energy_continuous.continuousOn
      (fun x hx r hr => by
        simpa [slope] using
          (curve.energy_hasDerivWithinAt x).liminf_right_slope_le hr)
      le_rfl
      hbound
      t
      ⟨hst, le_rfl⟩
  rw [gronwallBound_ε0] at hgronwall
  exact hgronwall

/-- A coercive inequality along a dissipation curve implies exponential decay
from time zero. -/

Open AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/SemigroupDecay.lean:79published source at 7bcd37294df1

Proof architecture

propagate a coercive energy-dissipation inequality exponentially between arbitrary starting and terminal times

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`simpa` closes the goal after a controlled simplification of a typed result.

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