Mathlib theorem · Existing formal mathematics
ProofAtlas

Existing in Mathlib · Analysis

Grönwall's Inequality

A local upper bound on the norm of a right derivative controls the function's norm everywhere on the interval through Mathlib's explicit Grönwall comparison function.

Exact theorem

Exact Mathlib statement

theorem norm_le_gronwallBound_of_norm_deriv_right_le {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {f f' : ℝ → E} {δ K ε : ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (ha : ‖f a‖ ≤ δ) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ K * ‖f x‖ + ε) : ∀ x ∈ Icc a b, ‖f x‖ ≤ gronwallBound δ K ε (x - a)

The theorem at a glance

Grönwall's inequality at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

Right-derivative control on [a,b), together with the initial norm bound, yields the recorded `gronwallBound` estimate throughout [a,b]. Explanatory diagram.
Detailed visual description

The theorem begins with ‖f a‖ ≤ δ, controls ‖f' x‖ by K·‖f x‖+ε at every right-differentiable point in [a,b), and concludes that ‖f x‖ stays below gronwallBound δ K ε (x−a) on the closed interval. The comparison function is δ+εt when K=0 and δ exp(Kt)+(ε/K)(exp(Kt)−1) otherwise. The checked bridge takes the norm, controls its lower limiting right slope, and invokes the scalar comparison theorem; no sign assumptions on K, δ, or ε are added.

Statement structure

From hypotheses to conclusion

Statement map for Grönwall's InequalityA pointwise right-derivative norm inequality produces an explicit Grönwall bound across the interval. The pinned upstream declaration is norm_le_gronwallBound_of_norm_deriv_right_le. The exact checked statement is theorem norm_le_gronwallBound_of_norm_deriv_right_le {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {f f' : ℝ → E} {δ K ε : ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (ha : ‖f a‖ ≤ δ) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ K * ‖f x‖ + ε) : ∀ x ∈ Icc a b, ‖f x‖ ≤ gronwallBound δ K ε (x - a).Mathematical readingA pointwiseright-derivative norminequality produces anexplicit Grönwall boundacross the interval.Pinned declarationmathlib ·norm_le_gronwallBound_of_norm_deriv_right_leExact checked formtheoremnorm_le_gronwallBound_of_norm_deriv_right_le{E : Type*}[NormedAddCommGroup E][NormedSpace ℝ E] {f f': ℝ → E} {δ K ε : ℝ} {ab : ℝ} (hf :ContinuousOn f (Icc ab)) (hf' : ∀ x ∈ Ico ab, HasDerivWithinAt f(f' x) (Ici x) x) (ha :‖f a‖ ≤ δ) (bound : ∀ x∈ Ico a b, ‖f' x‖ ≤ K *‖f x‖ + ε) : ∀ x ∈ Icc ab, ‖f x‖ ≤ gronwallBoundδ K ε (x - a)Statement map for Grönwall's InequalityA pointwise right-derivative norm inequality produces an explicit Grönwall bound across the interval. The pinned upstream declaration is norm_le_gronwallBound_of_norm_deriv_right_le. The exact checked statement is theorem norm_le_gronwallBound_of_norm_deriv_right_le {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {f f' : ℝ → E} {δ K ε : ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (ha : ‖f a‖ ≤ δ) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ K * ‖f x‖ + ε) : ∀ x ∈ Icc a b, ‖f x‖ ≤ gronwallBound δ K ε (x - a).Mathematical readingA pointwiseright-derivative norminequality produces anexplicit Grönwall boundacross the interval.Pinned declarationmathlib ·norm_le_gronwallBound_of_norm_deriv_right_leExact checked formtheoremnorm_le_gronwallBound_of_norm_deriv_right_le{E : Type*}[NormedAddCommGroup E][NormedSpace ℝ E] {f f': ℝ → E} {δ K ε : ℝ} {ab : ℝ} (hf :ContinuousOn f (Icc ab)) (hf' : ∀ x ∈ Ico ab, HasDerivWithinAt f(f' x) (Ici x) x) (ha :‖f a‖ ≤ δ) (bound : ∀ x∈ Ico a b, ‖f' x‖ ≤ K *‖f x‖ + ε) : ∀ x ∈ Icc ab, ‖f x‖ ≤ gronwallBoundδ K ε (x - a)

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

Vector-valued motion passes through its norm to a scalar path with three right-sided local slope cues, while the path remains under a comparison envelope on the recorded interval. Explanatory scientific diagram.
Detailed visual description

At left, a finite fan of directed fibers narrows through one circular lens. In the comparison field, one emerald curve starts with the gold envelope at the left boundary and stays strictly below it afterward through the common closed right endpoint. Exactly three cobalt one-sided tangent wedges touch the emerald curve and point only to the right. Beneath the field, the cobalt control rail has a closed left marker and an open right marker before the boundary; the gold conclusion rail has closed markers at both boundaries. The raster adds no monotonicity, sign, integral, or out-of-interval claim.

Why it matters

A mathematical landmark

Grönwall's inequality is one of the central comparison tools in analysis and differential equations. Mathlib's selected declaration exposes a precise norm-valued, one-sided-derivative formulation and a short source route that converts vector data into the preceding scalar comparison theorem.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally
Locally reproducedExact upstream declaration replayed
Reviewed pageCurrent public presentation reviewed
Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This page indexes Mathlib's norm-valued right-derivative form of Grönwall's inequality. A continuous f : ℝ → E on Icc a b, with right derivative f' x within Ici x for x in Ico a b, initial bound ‖f a‖ ≤ δ, and local bound ‖f' x‖ ≤ K * ‖f x‖ + ε, satisfies ‖f x‖ ≤ gronwallBound δ K ε (x - a) on Icc a b. The selected declaration does not state an integral form, add sign assumptions on K, δ, or ε, prove ODE existence or uniqueness, or cover points outside the recorded interval.

Source and local evidence

Where the theorem comes from

Existing declaration
norm_le_gronwallBound_of_norm_deriv_right_le in mathlib
Relationship
The checked artifact is the upstream declaration itself
Local Lean evidence
artifact.library.mathlib.gronwall-inequality.v001
Source
Open the pinned upstream reference