{
  "artifactId": "artifact.library.mathlib.picard-lindelof-theorem.v001",
  "candidateOnly": false,
  "category": "Analysis",
  "claimBoundary": "This page indexes Mathlib's differential-form local existence declaration for Picard–Lindelöf. In a complete real normed vector space, the bundled IsPicardLindelof hypotheses control the vector field on one closed interval and ball; for one initial point x in closedBall x₀ r, the theorem produces α with α t₀ = x and the recorded derivative-within relation on Icc tmin tmax. The selected endpoint does not assert uniqueness, maximality, global existence, continuation beyond that interval, or a flow for all initial points.",
  "collectionSlug": "landmark-theorems-in-mathlib",
  "exactFormalStatement": "theorem IsPicardLindelof.exists_eq_forall_mem_Icc_hasDerivWithinAt {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E → E} {tmin tmax : ℝ} {t₀ : Icc tmin tmax} {x₀ x : E} {a r L K : ℝ≥0} (hf : IsPicardLindelof f t₀ x₀ a r L K) (hx : x ∈ closedBall x₀ r) : ∃ α : ℝ → E, α t₀ = x ∧ ∀ t ∈ Icc tmin tmax, HasDerivWithinAt α (f t (α t)) (Icc tmin tmax) t",
  "family": "Ordinary differential equations",
  "id": "library.mathlib.picard-lindelof-theorem.v001",
  "links": {
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    "collectionData": "/data/collections/mathlib-landmarks.json",
    "entry": "/library-theorems/picard-lindelof-theorem/",
    "entryData": "/data/library-theorems/picard-lindelof-theorem.json",
    "evidence": "/proofs/artifact.library.mathlib.picard-lindelof-theorem.v001.html",
    "evidenceData": "/data/proofs/artifact.library.mathlib.picard-lindelof-theorem.v001.evidence.json",
    "source": "/sources/upstream/picard-lindelof-theorem/"
  },
  "landmark": {
    "completeFormalType": "theorem IsPicardLindelof.exists_eq_forall_mem_Icc_hasDerivWithinAt {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E → E} {tmin tmax : ℝ} {t₀ : Icc tmin tmax} {x₀ x : E} {a r L K : ℝ≥0} (hf : IsPicardLindelof f t₀ x₀ a r L K) (hx : x ∈ closedBall x₀ r) : ∃ α : ℝ → E, α t₀ = x ∧ ∀ t ∈ Icc tmin tmax, HasDerivWithinAt α (f t (α t)) (Icc tmin tmax) t",
    "concepts": [
      "ordinary differential equations",
      "local existence",
      "Banach spaces",
      "Picard iteration",
      "contraction mapping",
      "initial value problems"
    ],
    "plainLanguage": "Under the recorded local regularity and size conditions, every permitted initial point has a solution curve on the chosen closed time interval.",
    "poster": {
      "alt": "An ivory local-existence poster states the interval, ball, regularity, norm, and time-radius hypotheses above the existence of a curve through the initial point; candidate curves pass through a Picard integral funnel into one fixed solution curve aligned with the vector field.",
      "byteSize": 2531374,
      "caption": "The Picard operator on a complete space of admissible curves yields a fixed curve solving the initial-value equation on the recorded closed interval.",
      "description": "The poster leads with the complete-space and IsPicardLindelof setting, an initial point in the smaller closed ball, and the exact time-radius constraint. It then states the existence of a curve through x at t₀ whose derivative on the recorded interval follows f. Candidate curves narrow through the Picard integral operator to an emerald fixed curve, followed by three source-faithful movements: build the operator, find a contracting iterate, and differentiate the fixed integral equation. The footer keeps the selected endpoint at local existence and states that uniqueness is separate.",
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      ],
      "generatedAt": "2026-07-27T22:06:50-04:00",
      "height": 1536,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/picard-lindelof-theorem/theorem-poster-v2.png",
      "sha256": "sha256:5553f9b0b7b15a129faf4bbf0d8f36097b788a46afd6758f9444616dd5e8ddc8",
      "title": "Picard–Lindelöf local existence at a glance",
      "width": 1024,
      "kind": "theorem_poster",
      "transcript": "ANALYSIS · DIFFERENTIAL EQUATIONS\nPICARD–LINDELÖF\nLOCAL EXISTENCE IN A BANACH SPACE\nt₀ ∈ [tmin,tmax] · x ∈ closedBall(x₀,r)\nLipschitz in state · continuous in time · ‖f‖ ≤ L\nL · max(tmax−t₀, t₀−tmin) ≤ a−r\nTHERE EXISTS A CURVE α\nα(t₀) = x\ndα/dt = f(t, α(t)) on [tmin,tmax]\nHOW THE CHECKED ROUTE MOVES\n1 · Build the Picard integral operator on a complete space of curves\n2 · Show that some iterate is contracting\n3 · Obtain a fixed curve and differentiate its integral equation\nEXACT SCOPE\nLocal existence on the recorded closed interval. Uniqueness is separate."
    },
    "visual": {
      "alt": "Inside a dark engraved local domain, several ivory and cobalt curves share one initial point, pass through nested gold contraction rings with rapidly shrinking separation, and leave as one emerald fixed curve whose tangent follows the surrounding vector field.",
      "byteSize": 3165628,
      "caption": "Factorially shrinking Picard iterates lead to a contracting iterate and hence a fixed admissible curve on the local interval.",
      "description": "A bounded time corridor crosses a larger state-space ball and carries several admissible curves through one highlighted initial point. Concentric gold operator rings narrow successive cobalt curve bundles toward a single emerald fixed curve. The same curve continues through a local vector field with matching tangents, while the enclosing corridor keeps the conclusion local rather than global.",
      "derivatives": [
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          "height": 427,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/picard-lindelof-theorem/theorem-schematic-v1-640.webp",
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          "publicPath": "/assets/library-theorems/picard-lindelof-theorem/theorem-schematic-v1-1200.webp",
          "sha256": "sha256:99b2852377944369e21048c0f60974b87b3d1f72be9ff58387d1c9bbdaa0455e",
          "width": 1200
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      ],
      "generatedAt": "2026-07-27T10:19:03-04:00",
      "height": 1024,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/picard-lindelof-theorem/theorem-schematic-v1.png",
      "sha256": "sha256:31ed7403748bdb50c6233e86cbd978aed7f81ca1fbc3ee3b8985557e9aa49ea2",
      "title": "Picard iteration inside the local domain",
      "width": 1536
    },
    "walkthrough": null,
    "whyLandmark": "Picard–Lindelöf is the foundational local-existence mechanism for ordinary differential equations. Mathlib's route makes the functional-analytic engine explicit: factorial control makes an iterate of the Picard operator contracting, a fixed point supplies the integral equation, and differentiation gives the local solution."
  },
  "nonClaims": [
    "The selected declaration proves local existence for one permitted initial point; it does not include uniqueness.",
    "The conclusion is confined to the recorded closed interval and does not assert a maximal or global solution.",
    "The selected endpoint does not construct a flow for every initial point; stronger flow declarations appear separately later in the source.",
    "The result assumes the bundled IsPicardLindelof conditions, including completeness of E, local Lipschitz and continuity control, a norm bound, and a time-radius inequality.",
    "ProofAtlas is indexing an existing Mathlib theorem, and generated explanations and visuals are not proof evidence."
  ],
  "publicExportEligible": true,
  "relationship": "upstream_direct",
  "schemaVersion": "library-theorem-entry-data.v1",
  "statementId": "statement.library.mathlib.picard-lindelof-theorem.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": "Local Lipschitz and size control give a solution curve through a permitted initial point on the recorded closed interval.",
  "targetId": "target.library.mathlib.picard-lindelof-theorem.v001",
  "title": "Picard–Lindelöf Local Existence Theorem",
  "upstreamOrigin": {
    "declarationName": "IsPicardLindelof.exists_eq_forall_mem_Icc_hasDerivWithinAt",
    "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/Analysis/ODE/PicardLindelof.lean#L742",
    "packageName": "mathlib",
    "repositoryUrl": "https://github.com/leanprover-community/mathlib4",
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    "sourceFile": "Mathlib/Analysis/ODE/PicardLindelof.lean",
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  "publicPresentationReview": {
    "status": "reviewed"
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}