Language-integrated query is a popular and powerful programming construct allowing database queries and ordinary program code to interoperate seamlessly and safely. Language-integrated query techniques rely on classical results about the nested relational calculus stating that its queries can be algorithmically translated to SQL, as long as their result type is a flat relation. Cooper and others advocated higher-order nested relational calculi as a basis for language-integrated queries in functional languages such as Links and F#. However, the translation of higher-order relational queries to SQL relies on a rewrite system for which no strong normalization proof has been published: a previous proof attempt does not deal correctly with rewrite rules that duplicate subterms. This paper fills the gap in the literature, explaining the difficulty with the previous attempt, and showing how to extend the $toptop$-lifting approach of Lindley and Stark to accommodate duplicating rewrites. We also show how to extend the proof to a recently-introduced calculus for heterogeneous queries mixing set and multiset semantics.
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