Congratulations: We've made it to the end!

Looking Back

We've covered quite a bit of ground so far. Here's a quick review...

• Functional programming:
  • "declarative" programming style (recursion over immutable data structures, rather than looping over mutable arrays or pointer structures)
  • higher-order functions
  • polymorphism

• Logic, the mathematical basis for software engineering:

                 logic                        calculus
          --------------------   ~   ----------------------------
          software engineering       mechanical/civil engineering
  

  • inductively defined sets and relations
  • inductive proofs
  • proof objects

• Coq, an industrial-strength proof assistant
  • functional core language
  • core tactics
  • automation

Looking Forward

If what you've seen so far has whetted your interest, you have two choices for further reading in the Software Foundations series:

• Logic Foundations (volume 1, by a set of authors similar to this book's) continues on with this material to introduce even more concepts in Coq, leading up to some simple topics in programming language semantics.
• Programming Language Foundations (volume 2, by a set of authors similar to this book's) covers material that might be found in a graduate course on the theory of programming languages, including Hoare logic, operational semantics, and type systems.
• Verified Functional Algorithms (volume 3, by Andrew Appel) builds on the themes of functional programming and program verification in Coq, addressing a range of topics that might be found in a standard data structures course, with an eye to formal verification.

Other sources

Here are some other good places to learn more...

• Other textbooks on discrete mathematics abound, and they cover plenty of material we didn't discuss. There are also whole fields of study surround set theory or combinatorics---try taking such a class in the math department!
• For questions about Coq, the coq area of Stack Overflow (https://stackoverflow.com/questions/tagged/coq) is an excellent community resource. - Here are some great books on functional programming Learn You - a Haskell for Great Good, by Miran Lipovaca - [Lipovaca 2011]. Real World Haskell, by Bryan - O'Sullivan, John Goerzen, and Don Stewart - [O'Sullivan 2008] ...and many other excellent books - on Haskell, OCaml, Scheme, Racket, Scala, F sharp, etc., - etc. - And some further resources for Coq: Certified Programming - with Dependent Types, by Adam Chlipala \CITE{Chlipala - 2013}. Interactive Theorem Proving and Program - Development: Coq'Art: The Calculus of Inductive - Constructions, by Yves Bertot and Pierre Casteran - [Bertot 2004]. - If you're interested in real-world applications of formal verification to critical software, see the Postscript chapter of _Programming Language Foundations_. - For applications of Coq in building verified systems, the lectures and course materials for the 2017 DeepSpec Summer School are a great resource. https://deepspec.org/event/dsss17/index.html