We describe several views of the semantics of a simple programming language as formal documents in the calculus of inductive constructions that can be verified by the Coq proof system. Covered aspects are natural semantics, denotational semantics, axiomatic semantics, and abstract interpretation. Descriptions as recursive functions are also provided whenever suitable, thus yielding a a verification condition generator and a static analyser that can be run inside the theorem prover for use in reflective proofs. Extraction of an interpreter from the denotational semantics is also described. All different aspects are formally proved sound with respect to the natural semantics specification.

More work on mechanized metatheory with an eye towards naturalness of representation and automation. This seems to me to relate to Adam Chlipala's work on A Certified Type-Preserving Compiler from Lambda Calculus to Assembly Language, which relies on denotational semantics and proof by reflection, in crucial ways. More generally, it seems to reinforce an important trend in using type-theory-based theorem provers to tackle programming language design from the semantic point of view (see also A Very Modal Model of a Modern, Major, General Type System and Verifying Semantic Type Soundness of a Simple Compiler). I find the trend exciting, but of course I also wonder how far we can practically go with it today, given that the overwhelming majority of the literature, including our beloved Types and Programming Languages, is based on A Syntactic Approach to Type Soundness. Even the upcoming How to write your next POPL paper in Coq at POPL '08 centers on the syntactic approach.

Is the syntactic approach barking up the wrong tree, in the long term? The semantic approach? Both? Neither?