chewi_lemma_1_2_20 · Samplinglib

Plain-English statement

A differentiable scalar quantity whose growth rate never exceeds c times its current value stays below the matching exponential curve.

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.

Mathematical statement

If g'(t) <= c g(t) on [0,T], then g(t) <= g(0) exp(ct) for every t in [0,T].

Intuition

Multiplying by exp(-ct) removes the allowed exponential growth, leaving a nonincreasing comparison quantity.

Conditions

T is positive
g is differentiable
the differential inequality holds at every point of the closed interval
the evaluation time belongs to the interval

Why these conditions cannot be dropped

differentiability supplies the one-sided slope required by the rigorous comparison theorem
the interval hypotheses delimit exactly where the differential inequality is known
positivity of T matches the source statement although the proof also handles degenerate intervals

Proof route

turn ordinary differentiability into Mathlib's right-slope hypothesis
apply the one-sided Gronwall bound with zero forcing and initial value g(0)
simplify the closed-form Gronwall solution and the time shift from zero

Lean interface notes

Icc 0 T represents the source interval
the proof reuses le_gronwallBound_of_liminf_deriv_right_le rather than reproving the comparison argument
this isolated source route is compiled even though its semigroup consumers remain partial

Lean learning studio · mathematics → formal proof

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Lean statement

theorem chewi_lemma_1_2_20
    {T c : ℝ} (_hT : 0 < T) (g : ℝ → ℝ)
    (hg : Differentiable ℝ g)
    (hbound : ∀ t ∈ Icc (0 : ℝ) T, deriv g t ≤ c * g t)
    {t : ℝ} (ht : t ∈ Icc (0 : ℝ) T) :
    g t ≤ g 0 * Real.exp (c * t) := by
  have hgronwall :=
    le_gronwallBound_of_liminf_deriv_right_le
      (f := g)
      (f' := deriv g)
      (δ := g 0)
      (K := c)
      (ε := 0)
      (a := 0)
      (b := T)
      hg.continuous.continuousOn
      (fun x _ r hr =>
        (hg x).hasDerivAt.hasDerivWithinAt.liminf_right_slope_le hr)
      le_rfl
      (fun x hx => by
        simpa using hbound x ⟨hx.1, le_of_lt hx.2⟩)
      t
      ht
  rw [gronwallBound_ε0, sub_zero] at hgronwall
  exact hgronwall

/-- A scalar energy/dissipation curve with an exact right-derivative identity.

`scale` records the coefficient in
`d/dt energy(t) = -scale * dissipation(t)`. The derivative is taken within
`[t, ∞)`, matching semigroups defined by nonnegative time increments. -/

Open AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/SemigroupDecay.lean:38published source at 7bcd37294df1