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However, the foundations of these techniques are lacking: although SQL, the most widely-used database query language, supports <jats:italic>heterogeneous<\/jats:italic> queries that mix set and multiset semantics, these important capabilities are not supported by known correctness results or implementations that assume <jats:italic>homogeneous<\/jats:italic> collections. In this paper we study language-integrated query for a heterogeneous query language <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal {NRC}_{\\lambda }( Set,Bag )$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>NRC<\/mml:mi>\n                      <mml:mi>\u03bb<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>S<\/mml:mi>\n                      <mml:mi>e<\/mml:mi>\n                      <mml:mi>t<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mi>B<\/mml:mi>\n                      <mml:mi>a<\/mml:mi>\n                      <mml:mi>g<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> that combines set and multiset constructs. We show how to normalize and translate queries to SQL, and develop a novel approach to querying heterogeneous nested collections, based on the insight that \u201clocal\u201d query subexpressions that calculate nested subcollections can be \u201clifted\u201d to the top level analogously to lambda-lifting for local function definitions.<\/jats:p>","DOI":"10.1007\/978-3-030-72019-3_21","type":"book-chapter","created":{"date-parts":[[2021,3,22]],"date-time":"2021-03-22T14:03:10Z","timestamp":1616421790000},"page":"579-606","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Query Lifting"],"prefix":"10.1007","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2361-8538","authenticated-orcid":false,"given":"Wilmer","family":"Ricciotti","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1307-9286","authenticated-orcid":false,"given":"James","family":"Cheney","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,3,23]]},"reference":[{"key":"21_CR1","doi-asserted-by":"crossref","unstructured":"Benedikt, M., Pradic, P.: Generating collection transformations from proofs. 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