{ "artifactId": "artifact.library.mathlib.liouville-theorem.v001", "candidateOnly": false, "category": "Complex analysis", "claimBoundary": "This target indexes Mathlib's complex Fréchet-differentiable normed-space theorem: E and F are complex normed spaces, f is differentiable on all of E, and its range is bounded. The conclusion is exact equality with a constant function. The declaration is more general than the scalar ℂ→ℂ slogan and does not require F to be complete.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "theorem Differentiable.exists_eq_const_of_bounded {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type v} [NormedAddCommGroup F] [NormedSpace ℂ F] {f : E → F} (hf : Differentiable ℂ f) (hb : IsBounded (Set.range f)) : ∃ c : F, f = Function.const E c", "family": "Entire functions", "id": "library.mathlib.liouville-theorem.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/liouville-theorem/", "entryData": "/data/library-theorems/liouville-theorem.json", "evidence": "/proofs/artifact.library.mathlib.liouville-theorem.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.liouville-theorem.v001.evidence.json", "source": "/sources/upstream/liouville-theorem/" }, "landmark": { "completeFormalType": "theorem Differentiable.exists_eq_const_of_bounded {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type v} [NormedAddCommGroup F] [NormedSpace ℂ F] {f : E → F} (hf : Differentiable ℂ f) (hb : IsBounded (Set.range f)) : ∃ c : F, f = Function.const E c", "concepts": [ "complex differentiability", "bounded range", "entire map", "constant function", "complex normed spaces" ], "plainLanguage": "Let E and F be complex normed vector spaces, and let f map all of E into F. If f is complex differentiable everywhere and its entire range stays bounded, then every input has exactly the same output.", "poster": { "alt": "An ivory complex-analysis poster shows an everywhere complex-differentiable map from complex normed space E to F, a bounded range, and the conclusion that f equals one constant map.", "byteSize": 2332998, "caption": "A bounded entire map between complex normed spaces has no room to vary.", "description": "The poster separates the unbounded input space, global differentiability, bounded output range, and constant conclusion. A Cauchy-estimate route visually lets the derivative bound shrink as radius grows, without restricting the headline theorem to scalar functions.", "derivatives": [ { "byteSize": 95172, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/liouville-theorem/theorem-poster-v1-640.webp", "sha256": "sha256:a0ae2f2ea72eb8931f71c56dc5b3caa59fa706125aa8eaf506e5b17143044c90", "width": 640 }, { "byteSize": 187520, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/liouville-theorem/theorem-poster-v1-1024.webp", "sha256": "sha256:c0884d7e42c4333724be49f89b6b9377f0ca0fcd1774bf93c2fbd42ccb5f697c", "width": 1024 } ], "generatedAt": "2026-07-25T06:08:46.113Z", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/liouville-theorem/theorem-poster-v1.png", "sha256": "sha256:f124af5592e492f15e0b517b0a64d82a4aa6af45d1b78d8dd2db552b9657d565", "title": "Liouville's theorem at a glance", "width": 1024, "kind": "theorem_poster", "transcript": "COMPLEX ANALYSIS · GLOBAL RIGIDITY\nLIOUVILLE'S THEOREM\nf : E → F\ncomplex differentiable everywhere · range(f) bounded\nthere exists c ∈ F such that f = const c\nBOUNDED ENTIRE MAPS CANNOT VARY\nHOW THE CHECKED ROUTE MOVES\n1 · Restrict f to each complex line through two points\n2 · Apply Cauchy's derivative estimate on expanding circles\n3 · Let the radius grow and force the derivative to vanish\n4 · Conclude every pair of function values is equal\nEXACT SCOPE\nArbitrary complex normed spaces E and F; no completeness assumption on F." }, "visual": { "alt": "An unbounded complex normed domain sends many smooth analytic paths into a bounded target region, where every image is forced onto one distinguished point.", "byteSize": 2324342, "caption": "Global complex differentiability plus bounded range forces the entire map to be constant.", "description": "A broad, unbounded domain E carries many sample points and complex directions. Their smooth map paths enter one bounded contour in F and terminate at the same point c, emphasizing normed-space generality rather than only a complex-plane graph.", "derivatives": [ { "byteSize": 41060, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/liouville-theorem/theorem-schematic-v1-640.webp", "sha256": "sha256:cbb3388ad59583af41f47080f8949f229a1df6888fce6eb09632f90b623198f6", "width": 640 }, { "byteSize": 116346, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/liouville-theorem/theorem-schematic-v1-1200.webp", "sha256": "sha256:1be83772ccdf74b9b12be52ac49bed908dcc513659fe29fa0e430dd89f621c90", "width": 1200 } ], "generatedAt": "2026-07-25T06:08:46.113Z", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/liouville-theorem/theorem-schematic-v1.png", "sha256": "sha256:714dde72f2f1f2247673ed29ad782aa03fcc5dcdde70691f751b5286e8acd26c", "title": "Liouville's theorem schematic", "width": 1536 }, "walkthrough": null, "whyLandmark": "Liouville's theorem converts a global growth restriction into complete rigidity. It is a central tool in complex analysis and a standard bridge to the Fundamental Theorem of Algebra." }, "nonClaims": [ "Proof Atlas did not originate Liouville's theorem or Mathlib's declaration.", "The selected endpoint is not restricted to scalar-valued functions on the complex plane.", "The theorem does not apply to a function assumed differentiable only on a proper bounded domain.", "This is not Liouville's number-theoretic approximation theorem.", "The generated explanation and visuals are not proof evidence." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.liouville-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": "A bounded, everywhere complex-differentiable map between complex normed spaces is constant.", "targetId": "target.library.mathlib.liouville-theorem.v001", "title": "Liouville's Theorem", "upstreamOrigin": { "declarationName": "Differentiable.exists_eq_const_of_bounded", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/Analysis/Complex/Liouville.lean#L128", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:de49ced84f5799222893a523667144dd01cfa0b793cf1b2deb72268b3651700d", "sourceByteLength": 7933, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/Analysis/Complex/Liouville.lean", "sourceLine": 128, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }