Polymorphic Rewriting Conserves Algebraic Confluence

We study combinations of many-sorted algebraic term rewriting systems and polymorphic lambda term rewriting. Algebraic and lambda terms are mixed by adding the symbols of the algebraic signature to the polymorphic lambda calculus, as higher-order constants. We show that if a many-sorted algebraic rewrite system R has the Church-Rosser property (is confluent), then R + β + type-β + type-η rewriting of mixed terms has the Church-Rosser property too. η reduction does not commute with algebraic reduction, in general. However, using long normal forms, we show that if R is canonical (confluent and strongly normalizing) then equational provability from R + β + η + type-β + type-η is still decidable.