We introduce System F C , which extends System F with support for non-syntactic type equality. There are two main extensions: (i) explicit witnesses for type equalities, and (ii) open, non-parametric type functions, given meaning by toplevel equality axioms. Unlike System F, F C is expressive enough to serve as a target for several different source-language features, including Haskell's newtype, generalised algebraic data types, associated types, functional dependencies, and perhaps more besides. System F with Type Equality Coercions AbstractWe introduce System FC, which extends System F with support for non-syntactic type equality. There are two main extensions: (i) explicit witnesses for type equalities, and (ii) open, non-parametric type functions, given meaning by top-level equality axioms. Unlike System F, FC is expressive enough to serve as a target for several different source-language features, including Haskell's newtype, generalised algebraic data types, associated types, functional dependencies, and perhaps more besides.

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The prime number theorem, established by Hadamard and de la Vallée Poussin independently in 1896, asserts that the density of primes in the positive integers is asymptotic to 1/ ln x. Whereas their proofs made serious use of the methods of complex analysis, elementary proofs were provided by Selberg and Erdös in 1948. We describe a formally verified version of Selberg's proof, obtained using the Isabelle proof assistant.

Despite the emerging use of diamond-like carbon (DLC) as a coating for medical devices, few studies have examined the resistance of DLC coatings onto medical polymers to both microbial adherence and encrustation. In this study, amorphous DLC of a range of refractive indexes (1.7-1.9) and thicknesses (100-600 nm) was deposited onto polyurethane, a model polymer, and the resistance to microbial adherence (Escherichia coli; clinical isolate) and encrustation examined using in vitro models. In comparison to the native polymer, the advancing and receding contact angles of DLC-coated polyurethane were lower, indicating greater hydrophilic properties. No relationship was observed between refractive index, thickness, and advancing contact angle, as determined using multiple correlation analysis. The resistances of the various DLC-coated polyurethane films to encrustation and microbial adherence were significantly greater than that to polyurethane; however, there were individual differences between the resistances of the various DLC coatings. In general, increasing the refractive index of the coatings (100 nm thickness) decreased the resistance of the films to both hydroxyapatite and struvite encrustation and to microbial adherence. Films of lower thicknesses (100 and 200 nm; of defined refractive index, 1.8), exhibited the greatest resistance to encrustation and to microbial adherence. In conclusion, this study has uniquely illustrated both the microbial antiadherence properties and resistance to urinary encrustation of DLC-coated polyurethane. The resistances to encrustation and microbial adherence were substantial, and in light of this, it is suggested that DLC coatings of low thickness and refractive index show particular promise as coatings of polymeric medical devices.