{ "artifactId": "artifact.library.mathlib.los-theorem.v001", "candidateOnly": false, "category": "Logic and foundations", "claimBoundary": "This target indexes Mathlib's sentence-level form of Łoś's theorem. For an ultrafilter u and a family M of nonempty L-structures, the ultraproduct satisfies a sentence φ exactly when M a satisfies φ for u-almost every a. “u-almost every” means membership in the ultrafilter; it is not numerical majority, probability, or measure. The selected endpoint does not itself display parameters for formulas with free variables.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "theorem FirstOrder.Language.Ultraproduct.sentence_realize {α : Type*} {M : α → Type*} {u : Ultrafilter α} {L : FirstOrder.Language.{u, v}} [∀ a, L.Structure (M a)] [∀ a : α, Nonempty (M a)] (φ : L.Sentence) : (u : Filter α).Product M ⊨ φ ↔ ∀ᶠ a : α in u, M a ⊨ φ", "family": "Ultraproducts and transfer", "id": "library.mathlib.los-theorem.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/los-theorem/", "entryData": "/data/library-theorems/los-theorem.json", "evidence": "/proofs/artifact.library.mathlib.los-theorem.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.los-theorem.v001.evidence.json", "source": "/sources/upstream/los-theorem/" }, "landmark": { "completeFormalType": "theorem FirstOrder.Language.Ultraproduct.sentence_realize {α : Type*} {M : α → Type*} {u : Ultrafilter α} {L : FirstOrder.Language.{u, v}} [∀ a, L.Structure (M a)] [∀ a : α, Nonempty (M a)] (φ : L.Sentence) : (u : Filter α).Product M ⊨ φ ↔ ∀ᶠ a : α in u, M a ⊨ φ", "concepts": [ "first-order language", "ultrafilter", "ultraproduct", "sentence satisfaction", "model-theoretic transfer" ], "plainLanguage": "Build an ultraproduct from a family of nonempty structures and an ultrafilter on their indices. A first-order sentence is true in the resulting ultraproduct exactly when the set of indices whose structures satisfy that sentence belongs to the ultrafilter.", "poster": { "alt": "An ivory and forest-green theorem poster states that a sentence is true in the ultraproduct exactly when it is true in ultrafilter-many factors, above a family of structures feeding one quotient construction and a two-way truth seam.", "byteSize": 2791993, "caption": "Sentence truth in an ultraproduct is equivalent to truth on an ultrafilter-large set of factors.", "description": "The poster keeps the exact sentence-level endpoint prominent. Every displayed factor feeds the construction, a gold contour distinguishes an ultrafilter-large satisfying region without implying a vote, and a balanced two-way mark communicates the if-and-only-if truth transfer.", "derivatives": [ { "byteSize": 78356, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/los-theorem/theorem-poster-v1-640.webp", "sha256": "sha256:cc6fefb7658b91f01c13da42ff617076caf2197145750a2d5f682ffa880a720b", "width": 640 }, { "byteSize": 205952, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/los-theorem/theorem-poster-v1-1024.webp", "sha256": "sha256:388124d10a6f6fb8f312f7c01c0ca917b43078eeeb0e7202de00a901bba300f6", "width": 1024 } ], "generatedAt": "2026-07-25T20:26:02Z", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/los-theorem/theorem-poster-v1.png", "sha256": "sha256:c9b838b27b7668d52016dc2f6100f67f13f3ca1949c3e7d2e81f836aefd58475", "title": "Łoś's theorem at a glance", "width": 1024, "kind": "theorem_poster", "transcript": "MODEL THEORY\nŁOŚ'S THEOREM\nA SENTENCE IS TRUE\nIN THE ULTRAPRODUCT\nIF AND ONLY IF\nIT IS TRUE IN\nULTRAFILTER-MANY FACTORS\nSENTENCES · NONEMPTY FACTORS" }, "visual": { "alt": "A large indexed family of abstract structures, including dim structures outside a gold ultrafilter-large region, sends every strand through a quotient ring into one illuminated ultraproduct while a separate two-way gold seam pairs the two truth states.", "byteSize": 3263138, "caption": "The full family forms the ultraproduct, while sentence truth is detected by the ultrafilter-large set of satisfying factors.", "description": "Distinct small structures fill the left field. A gold contour highlights the factors satisfying one sentence on an ultrafilter-large set, yet fine connections from both highlighted and unhighlighted factors enter the full product braid. A central quotient ring yields one coherent ultraproduct, and a separate two-way seam expresses the truth equivalence.", "derivatives": [ { "byteSize": 51748, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/los-theorem/theorem-schematic-v1-640.webp", "sha256": "sha256:8415401a244b618e5eafeaa78e3c580014cd8b5322d7b09a60da4a5ac64dcae7", "width": 640 }, { "byteSize": 219070, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/los-theorem/theorem-schematic-v1-1200.webp", "sha256": "sha256:3a247abb47aa9f928f31c5abfe26519b2f8a3a3f72d57eb81a921492cd9da629", "width": 1200 } ], "generatedAt": "2026-07-25T20:26:02Z", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/los-theorem/theorem-schematic-v1.png", "sha256": "sha256:99a44d5049c6590a9fc5e7e97d2d44793a5ea30407266065db93a2ca4c68e888", "title": "Łoś's theorem schematic", "width": 1536 }, "walkthrough": null, "whyLandmark": "Łoś's theorem is the central transfer principle for ultraproducts: first-order truth in the quotient structure is governed exactly by ultrafilter-large truth among the factors. It is foundational to compactness arguments, nonstandard models, and modern model theory." }, "nonClaims": [ "Proof Atlas did not originate Łoś's theorem or Mathlib's declaration.", "The theorem does not require every factor to satisfy the sentence; it requires the satisfying indices to belong to the ultrafilter.", "The selected endpoint is for sentences and does not directly state the more general formula-with-parameters result used in its proof.", "The theorem does not identify ultrafilter membership with finite majority, probability, or measure.", "The generated explanation and visuals are not proof evidence." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.los-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": "An ultraproduct satisfies a first-order sentence if and only if that sentence holds in ultrafilter-many factors.", "targetId": "target.library.mathlib.los-theorem.v001", "title": "Łoś's Theorem", "upstreamOrigin": { "declarationName": "FirstOrder.Language.Ultraproduct.sentence_realize", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/ModelTheory/Ultraproducts.lean#L154", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:ba32a045647e55dee5bc5b4534ede125eb6cc7bef523aec77dea5e980dfacd54", "sourceByteLength": 6337, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/ModelTheory/Ultraproducts.lean", "sourceLine": 154, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }