{
  "artifactId": "artifact.library.mathlib.gronwall-inequality.v001",
  "candidateOnly": false,
  "category": "Analysis",
  "claimBoundary": "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.",
  "collectionSlug": "landmark-theorems-in-mathlib",
  "exactFormalStatement": "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)",
  "family": "Differential inequalities and ODEs",
  "id": "library.mathlib.gronwall-inequality.v001",
  "links": {
    "collection": "/collections/landmark-theorems-in-mathlib/",
    "collectionData": "/data/collections/mathlib-landmarks.json",
    "entry": "/library-theorems/gronwall-inequality/",
    "entryData": "/data/library-theorems/gronwall-inequality.json",
    "evidence": "/proofs/artifact.library.mathlib.gronwall-inequality.v001.html",
    "evidenceData": "/data/proofs/artifact.library.mathlib.gronwall-inequality.v001.evidence.json",
    "source": "/sources/upstream/gronwall-inequality/"
  },
  "landmark": {
    "completeFormalType": "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)",
    "concepts": [
      "Grönwall inequality",
      "differential inequalities",
      "right derivatives",
      "normed vector spaces",
      "comparison functions",
      "ordinary differential equations"
    ],
    "plainLanguage": "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.",
    "poster": {
      "alt": "An ivory Grönwall poster places a green norm trajectory beneath a gold comparison envelope from a to b. Blue one-sided tangent arrows mark the half-open derivative-control interval, and three cards state the initial bound, local derivative inequality, and global gronwallBound conclusion above the exact two-case comparison function.",
      "byteSize": 2559750,
      "caption": "Right-derivative control on [a,b), together with the initial norm bound, yields the recorded `gronwallBound` estimate throughout [a,b].",
      "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.",
      "derivatives": [
        {
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          "height": 960,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/gronwall-inequality/theorem-poster-v2-640.webp",
          "sha256": "sha256:9cb45b0ea205ba62ffb457fc54607bbc11ef152d66930ea2986e4e931cab28a5",
          "width": 640
        },
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          "height": 1536,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/gronwall-inequality/theorem-poster-v2-1024.webp",
          "sha256": "sha256:756304e6778616487e89220009ae2ec0ffe2d77c2b0a46954afad5ce51595274",
          "width": 1024
        }
      ],
      "generatedAt": "2026-07-27T21:35:24Z",
      "height": 1536,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/gronwall-inequality/theorem-poster-v2.png",
      "sha256": "sha256:a70b65942d04388d02c9ed7fae9f03086beb75bc822820a4f0a494a76301e6a5",
      "title": "Grönwall's inequality at a glance",
      "width": 1024,
      "kind": "theorem_poster",
      "transcript": "ANALYSIS · DIFFERENTIAL INEQUALITIES\nGRÖNWALL'S INEQUALITY\nLOCAL RIGHT-DERIVATIVE CONTROL → GLOBAL NORM BOUND\na\nb\nSTART\n‖f a‖ ≤ δ\nCONTROL ON [a,b)\n‖f' x‖ ≤ K · ‖f x‖ + ε\nCONCLUSION ON [a,b]\n‖f x‖ ≤ gronwallBound δ K ε (x − a)\nTHE COMPARISON FUNCTION\nK = 0: δ + εt\nK ≠ 0: δ exp(Kt) + (ε/K)(exp(Kt) − 1)\nCHECKED ROUTE\nTake norms · control the lower right slope · apply scalar comparison\nEXACT SCOPE\nContinuous f, right derivatives on [a,b), and the recorded norm inequality. No sign assumptions on K, δ, or ε."
    },
    "visual": {
      "alt": "A dark engraved vector bundle contracts through a norm lens into a green scalar curve. Eight right-facing cobalt tangent wedges track the half-open control interval, while a nonmonotone gold comparison envelope remains above the curve through the closed endpoint.",
      "byteSize": 3106870,
      "caption": "The vector-valued trajectory passes through its norm, local one-sided slope control, and a global comparison envelope on the recorded interval.",
      "description": "Vector-valued fibers at the left are reduced through a norm lens to one scalar-height path. Repeated right-facing tangent wedges encode derivatives within the right half-line at interior points, and the green path remains below a distinct gold comparison envelope. An open marker ends the derivative-control strip before b while a closed marker retains the final estimate at b; the unlabeled curves avoid adding monotonicity or sign assumptions absent from the theorem.",
      "derivatives": [
        {
          "byteSize": 44994,
          "height": 427,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/gronwall-inequality/theorem-schematic-v1-640.webp",
          "sha256": "sha256:2d650efa6773d2803443dce614c204e5f56cc80d688f72febcec4e62319a73d8",
          "width": 640
        },
        {
          "byteSize": 190838,
          "height": 800,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/gronwall-inequality/theorem-schematic-v1-1200.webp",
          "sha256": "sha256:0f879c7299032bf4480eaa64307ddc717fac7f02f02db89707facbcbc20db51d",
          "width": 1200
        }
      ],
      "generatedAt": "2026-07-27T10:47:05-04:00",
      "height": 1024,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/gronwall-inequality/theorem-schematic-v1.png",
      "sha256": "sha256:8a5211f5355ded28f90e800107d8e211a8686a4ae890b645a12a8f8a38c97c5a",
      "title": "From one-sided slope control to a global envelope",
      "width": 1536
    },
    "walkthrough": null,
    "whyLandmark": "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."
  },
  "nonClaims": [
    "ProofAtlas did not originate Grönwall's inequality or Mathlib's declaration.",
    "The selected endpoint is a norm bound derived from right derivatives on Ico a b; it is not an unrestricted integral-form Grönwall theorem.",
    "The declaration does not assume K, δ, or ε are nonnegative.",
    "It does not itself prove existence or uniqueness for an ordinary differential equation.",
    "The estimate is asserted on Icc a b only, with the derivative hypothesis on Ico a b.",
    "Generated explanations and visuals are not proof evidence."
  ],
  "publicExportEligible": true,
  "relationship": "upstream_direct",
  "schemaVersion": "library-theorem-entry-data.v1",
  "statementId": "statement.library.mathlib.gronwall-inequality.v001",
  "statusAxes": {
    "acceptedAtlasResult": "Not recorded for the preferred artifact",
    "locallyReproduced": "Exact upstream declaration replayed",
    "reviewedPage": "Current public presentation reviewed",
    "upstreamIndexed": "Pinned source bytes verified locally"
  },
  "summary": "A pointwise right-derivative norm inequality produces an explicit Grönwall bound across the interval.",
  "targetId": "target.library.mathlib.gronwall-inequality.v001",
  "title": "Grönwall's Inequality",
  "upstreamOrigin": {
    "declarationName": "norm_le_gronwallBound_of_norm_deriv_right_le",
    "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/Analysis/ODE/Gronwall.lean#L135",
    "packageName": "mathlib",
    "repositoryUrl": "https://github.com/leanprover-community/mathlib4",
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    "sourceByteLength": 20753,
    "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6",
    "sourceFile": "Mathlib/Analysis/ODE/Gronwall.lean",
    "sourceLine": 135,
    "verificationKind": "git_worktree"
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  "publicPresentationReview": {
    "status": "reviewed"
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}