{ "artifactId": "artifact.library.mathlib.rice-theorem.v001", "candidateOnly": false, "category": "Computability and logic", "claimBoundary": "This target indexes Mathlib's implication form of Rice's theorem. For a semantic property C of partial functions, assume program-code membership through eval is computable. If one partial-recursive f belongs to C, then every partial-recursive g belongs to C. It is specific to Mathlib's partial-function and program-code model and does not classify arbitrary syntactic code properties or all noncomputable semantic properties.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "theorem ComputablePred.rice (C : Set (ℕ →. ℕ)) (h : ComputablePred fun c => Nat.Partrec.Code.eval c ∈ C) {f g} (hf : Nat.Partrec f) (hg : Nat.Partrec g) (fC : f ∈ C) : g ∈ C", "family": "Semantic properties of programs", "id": "library.mathlib.rice-theorem.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/rice-theorem/", "entryData": "/data/library-theorems/rice-theorem.json", "evidence": "/proofs/artifact.library.mathlib.rice-theorem.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.rice-theorem.v001.evidence.json", "source": "/sources/upstream/rice-theorem/" }, "landmark": { "completeFormalType": "theorem ComputablePred.rice (C : Set (ℕ →. ℕ)) (h : ComputablePred fun c => Nat.Partrec.Code.eval c ∈ C) {f g} (hf : Nat.Partrec f) (hg : Nat.Partrec g) (fC : f ∈ C) : g ∈ C", "concepts": [ "program code", "partial recursive function", "semantic property", "computable predicate", "semantic undecidability" ], "plainLanguage": "Suppose a semantic property of partial functions can be decided computably from program codes. If one partial-recursive function has that property, then every partial-recursive function has it.", "poster": { "alt": "An ivory, forest-green, and gold Rice theorem poster sends program codes through evaluation, shows partial-recursive f inside a semantic property, and uses a self-reference loop to force arbitrary partial-recursive g into the same property.", "byteSize": 2295845, "caption": "Computable semantic membership and one admitted partial-recursive witness force every partial-recursive function into the property.", "description": "Program-code plaques feed into behavioral traces. The exact implication is dominant: computable code membership, Nat.Partrec f, Nat.Partrec g, and f ∈ C lead through a self-referential code to g ∈ C. The poster explicitly excludes syntactic program properties.", "derivatives": [ { "byteSize": 113786, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/rice-theorem/theorem-poster-v2-640.webp", "sha256": "sha256:0474c19e2ec984c1d611143c0ce4a3d7e5cffc62d103f0112d001f6f963d3827", "width": 640 }, { "byteSize": 218610, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/rice-theorem/theorem-poster-v2-1024.webp", "sha256": "sha256:d1319bbda4b10e193f74d96ee5d54bfea181c141795f1cf40f58b17092b4a182", "width": 1024 } ], "generatedAt": "2026-07-25T17:18:00-04:00", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/rice-theorem/theorem-poster-v2.png", "sha256": "sha256:5d42f144f886ebe25517e0eb3a3c1d2bd4e1d22714ce882ae7e7b7c2f378452c", "title": "Rice's theorem at a glance", "width": 1024, "kind": "theorem_poster", "transcript": "RICE'S THEOREM\nDECIDABLE SEMANTIC PROPERTY\nC — A SEMANTIC PROPERTY OF PARTIAL FUNCTIONS\nPROGRAM-CODE MEMBERSHIP IN C IS COMPUTABLE\nNat.Partrec f\nNat.Partrec g\nf ∈ C → g ∈ C\nTHE CHECKED ROUTE\n1 · CHOOSE CODES FOR f AND g\n2 · BUILD A SELF-REFERENTIAL CODE\n3 · DECIDE WHETHER ITS BEHAVIOR LIES IN C\n4 · EITHER BRANCH FORCES g ∈ C\nEXACT SCOPE\nMATHLIB PARTIAL-RECURSIVE FUNCTIONS AND PROGRAM CODES\nNOT A CLAIM ABOUT SYNTACTIC PROGRAM PROPERTIES" }, "visual": { "alt": "Distinct geometric program-code plaques pass through an evaluation lens into partial-function traces, then a self-reference loop sends all displayed partial-recursive behaviors into one semantic-property region.", "byteSize": 2702564, "caption": "Once computable semantic membership admits one partial-recursive behavior, the fixed-point route forces every partial-recursive behavior into the same property.", "description": "Different program-code plaques feed one evaluation lens and emerge as behavioral traces. A central one-way self-reference loop leads to a single semantic-property region containing the admitted witness and receiving every other displayed partial-recursive behavior.", "derivatives": [ { "byteSize": 39940, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/rice-theorem/theorem-schematic-v2-640.webp", "sha256": "sha256:f4d37ced37929320492ae993dd8755cf1674b9d6cb514daaf7dbef519d2ae587", "width": 640 }, { "byteSize": 109632, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/rice-theorem/theorem-schematic-v2-1200.webp", "sha256": "sha256:fbb795bf21039ec1eaee7b81bd884cc5ee19a7d1df6c5a9875a9fec248e256c4", "width": 1200 } ], "generatedAt": "2026-07-25T17:22:00-04:00", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/rice-theorem/theorem-schematic-v2.png", "sha256": "sha256:3b7fa55ee8de3b299ee4b0b3a3e5aa609a0cc82c19de00b1bb3e166a19a9c5b8", "title": "Rice's theorem implication schematic", "width": 1536 }, "walkthrough": null, "whyLandmark": "Rice's theorem draws a foundational boundary around algorithmic reasoning about programs: a computable decision procedure cannot separate partial-recursive functions by a nontrivial semantic property. Mathlib's selected endpoint proves this through a self-referential code in its partial-evaluation model." }, "nonClaims": [ "Proof Atlas did not originate Rice's theorem or Mathlib's declaration.", "The formal statement is tied to Mathlib's `Nat.Partrec` and `Nat.Partrec.Code` representation, not every machine model simultaneously.", "The hypothesis decides a property of evaluated partial functions from their program codes; the theorem does not classify arbitrary syntactic properties of codes.", "The conclusion ranges over partial-recursive f and g, not all arbitrary partial functions.", "This endpoint is a one-way implication, not the separate rice₂ empty-or-universal equivalence.", "The generated explanation and visuals are not proof evidence." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.rice-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 computably decidable semantic property containing one partial-recursive function contains every partial-recursive function.", "targetId": "target.library.mathlib.rice-theorem.v001", "title": "Rice's Theorem", "upstreamOrigin": { "declarationName": "ComputablePred.rice", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/Computability/Halting.lean#L209", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:c2a073a05c631e7fc957577a66025e9ac36dac741f9aa865e0f053b17f0c85de", "sourceByteLength": 17535, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/Computability/Halting.lean", "sourceLine": 209, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }