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Welcome

This electronic book is a course on Software Foundations, the mathematical underpinnings of reliable software. Topics include basic concepts of logic, computer-assisted theorem proving, the Coq proof assistant, functional programming, operational semantics, Hoare logic, and static type systems. The exposition is intended for a broad range of readers, from advanced undergraduates to PhD students and researchers. No specific background in logic or programming languages is assumed, though a degree of mathematical maturity will be helpful. The principal novelty of the course is that it is one hundred percent formalized and machine-checked: the entire text is literally a script for Coq. It is intended to be read alongside an interactive session with Coq. All the details in the text are fully formalized in Coq, and the exercises are designed to be worked using Coq. The files are organized into a sequence of core chapters, covering about one semester's worth of material and organized into a coherent linear narrative, plus a number of "appendices" covering additional topics. All the core chapters are suitable for both upper-level undergraduate and graduate students.

Overview

Building reliable software is hard. The scale and complexity of modern systems, the number of people involved in building them, and the range of demands placed on them render it extremely difficult to build software that is even more-or-less correct, much less 100% correct. At the same time, the increasing degree to which information processing is woven into every aspect of society continually amplifies the cost of bugs and insecurities. Computer scientists and software engineers have responded to these challenges by developing a whole host of techniques for improving software reliability, ranging from recommendations about managing software projects and organizing programming teams (e.g., extreme programming) to design philosophies for libraries (e.g., model-view-controller, publish-subscribe, etc.) and programming languages (e.g., object-oriented programming, aspect-oriented programming, functional programming, ...) to mathematical techniques for specifying and reasoning about properties of software and tools for helping validate these properties. The present course is focused on this last set of techniques. The text weaves together five conceptual threads: (1) basic tools from logic for making and justifying precise claims about programs; (2) the use of proof assistants to construct rigorous logical arguments; (3) the idea of functional programming, both as a method of programming that simplifies reasoning about programs and as a bridge between programming and logic; (4) formal techniques for reasoning about the properties of specific programs (e.g., the fact that a sorting function or a compiler obeys some formal specification); and (5) the use of type systems for establishing well-behavedness guarantees for all programs in a given programming language (e.g., the fact that well-typed Java programs cannot be subverted at runtime). Each of these topics is easily rich enough to fill a whole course in its own right, so tackling all of them together naturally means that much will be left unsaid. Nevertheless, we hope readers will find that the themes illuminate and amplify each other and that bringing them together creates a foundation from which it will be easy to dig into any of them more deeply. Some suggestions for further reading can be found in the Postscript chapter. Bibliographic information for all cited works can be found in the Bib chapter.

Logic

Logic is the field of study whose subject matter is proofs — unassailable arguments for the truth of particular propositions. Volumes have been written about the central role of logic in computer science. Manna and Waldinger called it "the calculus of computer science," while Halpern et al.'s paper On the Unusual Effectiveness of Logic in Computer Science catalogs scores of ways in which logic offers critical tools and insights. Indeed, they observe that "As a matter of fact, logic has turned out to be significiantly more effective in computer science than it has been in mathematics. This is quite remarkable, especially since much of the impetus for the development of logic during the past one hundred years came from mathematics." In particular, the fundamental notion of inductive proofs is ubiquitous in all of computer science. You have surely seen them before, in contexts from discrete math to analysis of algorithms, but in this course we will examine them much more deeply than you have probably done so far.

Proof Assistants

The flow of ideas between logic and computer science has not been in just one direction: CS has also made important contributions to logic. One of these has been the development of software tools for helping construct proofs of logical propositions. These tools fall into two broad categories:

• Automated theorem provers provide "push-button" operation: you give them a proposition and they return either true, false, or ran out of time. Although their capabilities are limited to fairly specific sorts of reasoning, they have matured tremendously in recent years and are used now in a huge variety of settings. Examples of such tools include SAT solvers, SMT solvers, and model checkers.
• Proof assistants are hybrid tools that automate the more routine aspects of building proofs while depending on human guidance for more difficult aspects. Widely used proof assistants include Isabelle, Agda, Twelf, ACL2, PVS, and Coq, among many others.

This course is based around Coq, a proof assistant that has been under development, mostly in France, since 1983 and that in recent years has attracted a large community of users in both research and industry. Coq provides a rich environment for interactive development of machine-checked formal reasoning. The kernel of the Coq system is a simple proof-checker, which guarantees that only correct deduction steps are performed. On top of this kernel, the Coq environment provides high-level facilities for proof development, including powerful tactics for constructing complex proofs semi-automatically, and a large library of common definitions and lemmas. Coq has been a critical enabler for a huge variety of work across computer science and mathematics:

• As a platform for modeling programming languages, it has become a standard tool for researchers who need to describe and reason about complex language definitions. It has been used, for example, to check the security of the JavaCard platform, obtaining the highest level of common criteria certification, and for formal specifications of the x₈₆ and LLVM instruction sets and programming languages such as C.
• As an environment for developing formally certified software, Coq has been used, for example, to build CompCert, a fully-verified optimizing compiler for C, for proving the correctness of subtle algorithms involving floating point numbers, and as the basis for CertiCrypt, an environment for reasoning about the security of cryptographic algorithms.
• As a realistic environment for functional programming with dependent types, it has inspired numerous innovations. For example, the Ynot project at Harvard embedded "relational Hoare reasoning" (an extension of the Hoare Logic we will see later in this course) in Coq.
• As a proof assistant for higher-order logic, it has been used to validate a number of important results in mathematics. For example, its ability to include complex computations inside proofs made it possible to develop the first formally verified proof of the 4-color theorem. This proof had previously been controversial among mathematicians because part of it included checking a large number of configurations using a program. In the Coq formalization, everything is checked, including the correctness of the computational part. More recently, an even more massive effort led to a Coq formalization of the Feit-Thompson Theorem — the first major step in the classification of finite simple groups.

By the way, in case you're wondering about the name, here's what the official Coq web site says: "Some French computer scientists have a tradition of naming their software as animal species: Caml, Elan, Foc or Phox are examples of this tacit convention. In French, 'coq' means rooster, and it sounds like the initials of the Calculus of Constructions (CoC) on which it is based." The rooster is also the national symbol of France, and C-o-q are the first three letters of the name of Thierry Coquand, one of Coq's early developers.

Functional Programming

The term functional programming refers both to a collection of programming idioms that can be used in almost any programming language and to a family of programming languages designed to emphasize these idioms, including Haskell, OCaml, Standard ML, F#, Scala, Scheme, Racket, Common Lisp, Clojure, Erlang, and Coq. Functional programming has been developed over many decades — indeed, its roots go back to Church's lambda-calculus, which was invented in the 1930s, before there were even any computers! But since the early '90s it has enjoyed a surge of interest among industrial engineers and language designers, playing a key role in high-value systems at companies like Jane St. Capital, Microsoft, Facebook, and Ericsson. The most basic tenet of functional programming is that, as much as possible, computation should be pure, in the sense that the only effect of execution should be to produce a result: the computation should be free from side effects such as I/O, assignments to mutable variables, redirecting pointers, etc. For example, whereas an imperative sorting function might take a list of numbers and rearrange its pointers to put the list in order, a pure sorting function would take the original list and return a new list containing the same numbers in sorted order. One significant benefit of this style of programming is that it makes programs easier to understand and reason about. If every operation on a data structure yields a new data structure, leaving the old one intact, then there is no need to worry about how that structure is being shared and whether a change by one part of the program might break an invariant that another part of the program relies on. These considerations are particularly critical in concurrent programs, where every piece of mutable state that is shared between threads is a potential source of pernicious bugs. Indeed, a large part of the recent interest in functional programming in industry is due to its simpler behavior in the presence of concurrency. Another reason for the current excitement about functional programming is related to the first: functional programs are often much easier to parallelize than their imperative counterparts. If running a computation has no effect other than producing a result, then it does not matter where it is run. Similarly, if a data structure is never modified destructively, then it can be copied freely, across cores or across the network. Indeed, the "Map-Reduce" idiom, which lies at the heart of massively distributed query processors like Hadoop and is used by Google to index the entire web is a classic example of functional programming. For this course, functional programming has yet another significant attraction: it serves as a bridge between logic and computer science. Indeed, Coq itself can be viewed as a combination of a small but extremely expressive functional programming language plus with a set of tools for stating and proving logical assertions. Moreover, when we come to look more closely, we find that these two sides of Coq are actually aspects of the very same underlying machinery — i.e., proofs are programs.

Program Verification

Approximately the first third of the book is devoted to developing the conceptual framework of logic and functional programming and gaining enough fluency with Coq to use it for modeling and reasoning about nontrivial artifacts. From this point on, we increasingly turn our attention to two broad topics of critical importance to the enterprise of building reliable software (and hardware): techniques for proving specific properties of particular programs and for proving general properties of whole programming languages. For both of these, the first thing we need is a way of representing programs as mathematical objects, so we can talk about them precisely, together with ways of describing their behavior in terms of mathematical functions or relations. Our tools for these tasks are abstract syntax and operational semantics, a method of specifying programming languages by writing abstract interpreters. At the beginning, we work with operational semantics in the so-called "big-step" style, which leads to somewhat simpler and more readable definitions when it is applicable. Later on, we switch to a more detailed "small-step" style, which helps make some useful distinctions between different sorts of "nonterminating" program behaviors and is applicable to a broader range of language features, including concurrency. The first programming language we consider in detail is Imp, a tiny toy language capturing the core features of conventional imperative programming: variables, assignment, conditionals, and loops. We study two different ways of reasoning about the properties of Imp programs. First, we consider what it means to say that two Imp programs are equivalent in the intuitive sense that they yield the same behavior when started in any initial memory state. This notion of equivalence then becomes a criterion for judging the correctness of metaprograms — programs that manipulate other programs, such as compilers and optimizers. We build a simple optimizer for Imp and prove that it is correct. Second, we develop a methodology for proving that particular Imp programs satisfy formal specifications of their behavior. We introduce the notion of Hoare triples — Imp programs annotated with pre- and post-conditions describing what should be true about the memory in which they are started and what they promise to make true about the memory in which they terminate — and the reasoning principles of Hoare Logic, a "domain-specific logic" specialized for convenient compositional reasoning about imperative programs, with concepts like "loop invariant" built in. This part of the course is intended to give readers a taste of the key ideas and mathematical tools used in a wide variety of real-world software and hardware verification tasks.

Type Systems

Our final major topic, covering approximately the last third of the course, is type systems, a powerful set of tools for establishing properties of all programs in a given language. Type systems are the best established and most popular example of a highly successful class of formal verification techniques known as lightweight formal methods. These are reasoning techniques of modest power — modest enough that automatic checkers can be built into compilers, linkers, or program analyzers and thus be applied even by programmers unfamiliar with the underlying theories. Other examples of lightweight formal methods include hardware and software model checkers, contract checkers, and run-time property monitoring techniques for detecting when some component of a system is not behaving according to specification. This topic brings us full circle: the language whose properties we study in this part, the simply typed lambda-calculus, is essentially a simplified model of the core of Coq itself!

Further Reading

This text is intended to be self contained, but readers looking for a deeper treatment of a particular topic will find suggestions for further reading in the Postscript chapter.

Practicalities

System Requirements

Coq runs on Windows, Linux, and OS X. You will need:

• A current installation of Coq, available from the Coq home page. Everything should work with version 8.4 (or 8.5).
• An IDE for interacting with Coq. Currently, there are two choices:
  • Proof General is an Emacs-based IDE. It tends to be preferred by users who are already comfortable with Emacs. It requires a separate installation (google "Proof General").
  • CoqIDE is a simpler stand-alone IDE. It is distributed with Coq, so it should "just work" once you have Coq installed. It can also be compiled from scratch, but on some platforms this may involve installing additional packages for GUI libraries and such.

Exercises

Each chapter includes numerous exercises. Each is marked with a "star rating," which can be interpreted as follows:

• One star: easy exercises that underscore points in the text and that, for most readers, should take only a minute or two. Get in the habit of working these as you reach them.
• Two stars: straightforward exercises (five or ten minutes).
• Three stars: exercises requiring a bit of thought (ten minutes to half an hour).
• Four and five stars: more difficult exercises (half an hour and up).

Also, some exercises are marked "advanced", and some are marked "optional." Doing just the non-optional, non-advanced exercises should provide good coverage of the core material. Optional exercises provide a bit of extra practice with key concepts and introduce secondary themes that may be of interest to some readers. Advanced exercises are for readers who want an extra challenge (and, in return, a deeper contact with the material). Please do not post solutions to the exercises in any public place: Software Foundations is widely used both for self-study and for university courses. Having solutions easily available makes it much less useful for courses, which typically have graded homework assignments. The authors especially request that readers not post solutions to the exercises anyplace where they can be found by search engines.

Introduction

The functional programming style brings programming closer to simple, everyday mathematics: If a procedure or method has no side effects, then (ignoring efficiency) all we need to understand about it is how it maps inputs to outputs — that is, we can think of it as just a concrete method for computing a mathematical function. This is one sense of the word "functional" in "functional programming." The direct connection between programs and simple mathematical objects supports both formal correctness proofs and sound informal reasoning about program behavior. The other sense in which functional programming is "functional" is that it emphasizes the use of functions (or methods) as first-class values — i.e., values that can be passed as arguments to other functions, returned as results, included in data structures, etc. The recognition that functions can be treated as data in this way enables a host of useful and powerful idioms. Other common features of functional languages include algebraic data types and pattern matching, which make it easy to construct and manipulate rich data structures, and sophisticated polymorphic type systems supporting abstraction and code reuse. Coq shares all of these features. The first half of this chapter introduces the most essential elements of Coq's functional programming language. The second half introduces some basic tactics that can be used to prove simple properties of Coq programs.

Enumerated Types

One unusual aspect of Coq is that its set of built-in features is extremely small. For example, instead of providing the usual palette of atomic data types (booleans, integers, strings, etc.), Coq offers a powerful mechanism for defining new data types from scratch, from which all these familiar types arise as instances. Naturally, the Coq distribution comes with an extensive standard library providing definitions of booleans, numbers, and many common data structures like lists and hash tables. But there is nothing magic or primitive about these library definitions. To illustrate this, we will explicitly recapitulate all the definitions we need in this course, rather than just getting them implicitly from the library. To see how this definition mechanism works, let's start with a very simple example.

Days of the Week

The following declaration tells Coq that we are defining a new set of data values — a type.

The type is called day, and its members are monday, tuesday, etc. The second and following lines of the definition can be read "monday is a day, tuesday is a day, etc." Having defined day, we can write functions that operate on days.

One thing to note is that the argument and return types of this function are explicitly declared. Like most functional programming languages, Coq can often figure out these types for itself when they are not given explicitly — i.e., it performs type inference — but we'll include them to make reading easier. Having defined a function, we should check that it works on some examples. There are actually three different ways to do this in Coq. First, we can use the command Compute to evaluate a compound expression involving next_weekday.

(We show Coq's responses in comments, but, if you have a computer handy, this would be an excellent moment to fire up the Coq interpreter under your favorite IDE — either CoqIde or Proof General — and try this for yourself. Load this file, Basics.v, from the book's accompanying Coq sources, find the above example, submit it to Coq, and observe the result.) Second, we can record what we expect the result to be in the form of a Coq example:

This declaration does two things: it makes an assertion (that the second weekday after saturday is tuesday), and it gives the assertion a name that can be used to refer to it later. Having made the assertion, we can also ask Coq to verify it, like this:

The details are not important for now (we'll come back to them in a bit), but essentially this can be read as "The assertion we've just made can be proved by observing that both sides of the equality evaluate to the same thing, after some simplification." Third, we can ask Coq to extract, from our Definition, a program in some other, more conventional, programming language (OCaml, Scheme, or Haskell) with a high-performance compiler. This facility is very interesting, since it gives us a way to construct fully certified programs in mainstream languages. Indeed, this is one of the main uses for which Coq was developed. We'll come back to this topic in later chapters.

Booleans

In a similar way, we can define the standard type bool of booleans, with members true and false.

Although we are rolling our own booleans here for the sake of building up everything from scratch, Coq does, of course, provide a default implementation of the booleans in its standard library, together with a multitude of useful functions and lemmas. (Take a look at Coq.Init.Datatypes in the Coq library documentation if you're interested.) Whenever possible, we'll name our own definitions and theorems so that they exactly coincide with the ones in the standard library. Functions over booleans can be defined in the same way as above:

The last two illustrate Coq's syntax for multi-argument function definitions. The corresponding multi-argument application syntax is illustrated by the following four "unit tests," which constitute a complete specification — a truth table — for the orb function:

We can also introduce some familiar syntax for the boolean operations we have just defined. The Infix command defines new, infix notation for an existing definition.

A note on notation: In .v files, we use square brackets to delimit fragments of Coq code within comments; this convention, also used by the coqdoc documentation tool, keeps them visually separate from the surrounding text. In the html version of the files, these pieces of text appear in a different font. The special phrases Admitted and admit can be used as a placeholder for an incomplete definition or proof. We'll use them in exercises, to indicate the parts that we're leaving for you — i.e., your job is to replace admit or Admitted with real definitions or proofs.

Exercise: 1 star (nandb)

Remove admit and complete the definition of the following function; then make sure that the Example assertions below can each be verified by Coq. (Remove "Admitted." and fill in each proof, following the model of the orb tests above.) The function should return true if either or both of its inputs are false.

☐

Exercise: 1 star (andb3)

Do the same for the andb3 function below. This function should return true when all of its inputs are true, and false otherwise.

☐

Function Types

Every expression in Coq has a type, describing what sort of thing it computes. The Check command asks Coq to print the type of an expression. For example, the type of negb true is bool.

Functions like negb itself are also data values, just like true and false. Their types are called function types, and they are written with arrows.

The type of negb, written bool → bool and pronounced "bool arrow bool," can be read, "Given an input of type bool, this function produces an output of type bool." Similarly, the type of andb, written bool → bool → bool, can be read, "Given two inputs, both of type bool, this function produces an output of type bool."

Modules

Coq provides a module system, to aid in organizing large developments. In this course we won't need most of its features, but one is useful: If we enclose a collection of declarations between Module X and End X markers, then, in the remainder of the file after the End, these definitions are referred to by names like X.foo instead of just foo. Here, we use this feature to introduce the definition of the type nat in an inner module so that it does not interfere with the one from the standard library, which comes with a bit of special notational magic.

Numbers

The types we have defined so far are examples of "enumerated types": their definitions explicitly enumerate a finite set of elements. A more interesting way of defining a type is to give a collection of inductive rules describing its elements. For example, we can define the natural numbers as follows:

The clauses of this definition can be read:

• O is a natural number (note that this is the letter "O," not the numeral "0").
• S is a "constructor" that takes a natural number and yields another one — that is, if n is a natural number, then S n is too.

Let's look at this in a little more detail. Every inductively defined set (day, nat, bool, etc.) is actually a set of expressions. The definition of nat says how expressions in the set nat can be constructed:

• the expression O belongs to the set nat;
• if n is an expression belonging to the set nat, then S n is also an expression belonging to the set nat; and
• expressions formed in these two ways are the only ones belonging to the set nat.

The same rules apply for our definitions of day and bool. The annotations we used for their constructors are analogous to the one for the O constructor, indicating that they don't take any arguments. These three conditions are the precise force of the Inductive declaration. They imply that the expression O, the expression S O, the expression S (S O), the expression S (S (S O)), and so on all belong to the set nat, while other expressions like true, andb true false, and S (S false) do not. We can write simple functions that pattern match on natural numbers just as we did above — for example, the predecessor function:

The second branch can be read: "if n has the form S n' for some n', then return n'."

Because natural numbers are such a pervasive form of data, Coq provides a tiny bit of built-in magic for parsing and printing them: ordinary arabic numerals can be used as an alternative to the "unary" notation defined by the constructors S and O. Coq prints numbers in arabic form by default:

The constructor S has the type nat → nat, just like the functions minustwo and pred:

These are all things that can be applied to a number to yield a number. However, there is a fundamental difference between the first one and the other two: functions like pred and minustwo come with computation rules — e.g., the definition of pred says that pred 2 can be simplified to 1 — while the definition of S has no such behavior attached. Although it is like a function in the sense that it can be applied to an argument, it does not do anything at all! For most function definitions over numbers, just pattern matching is not enough: we also need recursion. For example, to check that a number n is even, we may need to recursively check whether n-2 is even. To write such functions, we use the keyword Fixpoint.

We can define oddb by a similar Fixpoint declaration, but here is a simpler definition that is a bit easier to work with:

(You will notice if you step through these proofs that simpl actually has no effect on the goal — all of the work is done by reflexivity. We'll see more about why that is shortly.) Naturally, we can also define multi-argument functions by recursion.

Adding three to two now gives us five, as we'd expect.

The simplification that Coq performs to reach this conclusion can be visualized as follows:

As a notational convenience, if two or more arguments have the same type, they can be written together. In the following definition, (n m : nat) means just the same as if we had written (n : nat) (m : nat).

You can match two expressions at once by putting a comma between them:

The _ in the first line is a wildcard pattern. Writing _ in a pattern is the same as writing some variable that doesn't get used on the right-hand side. This avoids the need to invent a bogus variable name.

Exercise: 1 star (factorial)

Recall the standard mathematical factorial function:

       factorial(0)  =  1
       factorial(n)  =  n * factorial(n-1)     (if n>0)

Translate this into Coq.

☐ We can make numerical expressions a little easier to read and write by introducing notations for addition, multiplication, and subtraction.

(The level, associativity, and nat_scope annotations control how these notations are treated by Coq's parser. The details are not important, but interested readers can refer to the optional "More on Notation" section at the end of this chapter.) Note that these do not change the definitions we've already made: they are simply instructions to the Coq parser to accept x + y in place of plus x y and, conversely, to the Coq pretty-printer to display plus x y as x + y. When we say that Coq comes with nothing built-in, we really mean it: even equality testing for numbers is a user-defined operation! The beq_nat function tests natural numbers for equality, yielding a boolean. Note the use of nested matches (we could also have used a simultaneous match, as we did in minus.)

The leb function tests whether its first argument is less than or equal to its second argument, yielding a boolean.

☐

Proof by Simplification

Now that we've defined a few datatypes and functions, let's turn to stating and proving properties of their behavior. Actually, we've already started doing this: each Example in the previous sections makes a precise claim about the behavior of some function on some particular inputs. The proofs of these claims were always the same: use simpl to simplify both sides of the equation, then use reflexivity to check that both sides contain identical values. The same sort of "proof by simplification" can be used to prove more interesting properties as well. For example, the fact that 0 is a "neutral element" for + on the left can be proved just by observing that 0 + n reduces to n no matter what n is, a fact that can be read directly off the definition of plus.

(You may notice that the above statement looks different in the .v file in your IDE than it does in the HTML rendition in your browser, if you are viewing both. In .v files, we write the ∀ universal quantifier using the reserved identifier "forall." When the .v files are converted to HTML, this gets transformed into an upside-down-A symbol.) This is a good place to mention that reflexivity is a bit more powerful than we have admitted. In the examples we have seen, the calls to simpl were actually not needed, because reflexivity can perform some simplification automatically when checking that two sides are equal; simpl was just added so that we could see the intermediate state — after simplification but before finishing the proof. Here is a shorter proof of the theorem:

Moreover, it will be useful later to know that reflexivity does somewhat more simplification than simpl does — for example, it tries "unfolding" defined terms, replacing them with their right-hand sides. The reason for this difference is that, if reflexivity succeeds, the whole goal is finished and we don't need to look at whatever expanded expressions reflexivity has created by all this simplification and unfolding; by contrast, simpl is used in situations where we may have to read and understand the new goal that it creates, so we would not want it blindly expanding definitions and leaving the goal in a messy state. The form of the theorem we just stated and its proof are almost exactly the same as the simpler examples we saw earlier; there are just a few differences. First, we've used the keyword Theorem instead of Example. This difference is purely a matter of style; the keywords Example and Theorem (and a few others, including Lemma, Fact, and Remark) mean exactly the same thing to Coq. Second, we've added the quantifier ∀ n:nat, so that our theorem talks about all natural numbers n. In order to prove theorems of this form, we need to to be able to reason by assuming the existence of an arbitrary natural number n. This is achieved in the proof by intros n, which moves the quantifier from the goal to a context of current assumptions. In effect, we start the proof by saying "Suppose n is some arbitrary number..." The keywords intros, simpl, and reflexivity are examples of tactics. A tactic is a command that is used between Proof and Qed to guide the process of checking some claim we are making. We will see several more tactics in the rest of this chapter and yet more in future chapters. Other similar theorems can be proved with the same pattern.

The _l suffix in the names of these theorems is pronounced "on the left." It is worth stepping through these proofs to observe how the context and the goal change. You may want to add calls to simpl before reflexivity to see the simplifications that Coq performs on the terms before checking that they are equal. Although simplification is powerful enough to prove some fairly general facts, there are many statements that cannot be handled by simplification alone. For instance, we cannot use it to prove that 0 is also a neutral element for + on the right.

(Can you explain why this happens? Step through both proofs with Coq and notice how the goal and context change.) When stuck in the middle of a proof, we can use the Abort command to give up on it for the moment.

The next chapter will introduce induction, a powerful technique that can be used for proving this goal. For the moment, though, let's look at a few more simple tactics.

Proof by Rewriting

This theorem is a bit more interesting than the others we've seen:

Instead of making a universal claim about all numbers n and m, it talks about a more specialized property that only holds when n = m. The arrow symbol is pronounced "implies." As before, we need to be able to reason by assuming the existence of some numbers n and m. We also need to assume the hypothesis n = m. The intros tactic will serve to move all three of these from the goal into assumptions in the current context. Since n and m are arbitrary numbers, we can't just use simplification to prove this theorem. Instead, we prove it by observing that, if we are assuming n = m, then we can replace n with m in the goal statement and obtain an equality with the same expression on both sides. The tactic that tells Coq to perform this replacement is called rewrite.

The first line of the proof moves the universally quantified variables n and m into the context. The second moves the hypothesis n = m into the context and gives it the name H. The third tells Coq to rewrite the current goal (n + n = m + m) by replacing the left side of the equality hypothesis H with the right side. (The arrow symbol in the rewrite has nothing to do with implication: it tells Coq to apply the rewrite from left to right. To rewrite from right to left, you can use rewrite ←. Try making this change in the above proof and see what difference it makes.)

Exercise: 1 star (plus_id_exercise)

Remove "Admitted." and fill in the proof.

☐ The Admitted command tells Coq that we want to skip trying to prove this theorem and just accept it as a given. This can be useful for developing longer proofs, since we can state subsidiary lemmas that we believe will be useful for making some larger argument, use Admitted to accept them on faith for the moment, and continue working on the main argument until we are sure it makes sense; then we can go back and fill in the proofs we skipped. Be careful, though: every time you say Admitted (or admit) you are leaving a door open for total nonsense to enter Coq's nice, rigorous, formally checked world! We can also use the rewrite tactic with a previously proved theorem instead of a hypothesis from the context. If the statement of the previously proved theorem involves quantified variables, as in the example below, Coq tries to instantiate them by matching with the current goal.

☐

Proof by Case Analysis

Of course, not everything can be proved by simple calculation and rewriting: In general, unknown, hypothetical values (arbitrary numbers, booleans, lists, etc.) can block simplification. For example, if we try to prove the following fact using the simpl tactic as above, we get stuck.

The reason for this is that the definitions of both beq_nat and + begin by performing a match on their first argument. But here, the first argument to + is the unknown number n and the argument to beq_nat is the compound expression n + 1; neither can be simplified. To make progress, we need to consider the possible forms of n separately. If n is O, then we can calculate the final result of beq_nat (n + 1) 0 and check that it is, indeed, false. And if n = S n' for some n', then, although we don't know exactly what number n + 1 yields, we can calculate that, at least, it will begin with one S, and this is enough to calculate that, again, beq_nat (n + 1) 0 will yield false. The tactic that tells Coq to consider, separately, the cases where n = O and where n = S n' is called destruct.

The destruct generates two subgoals, which we must then prove, separately, in order to get Coq to accept the theorem. The annotation "as [| n']" is called an intro pattern. It tells Coq what variable names to introduce in each subgoal. In general, what goes between the square brackets is a list of lists of names, separated by |. In this case, the first component is empty, since the O constructor is nullary (it doesn't have any arguments). The second component gives a single name, n', since S is a unary constructor. The - signs on the second and third lines are called bullets, and they mark the parts of the proof that correspond to each generated subgoal. The proof script that comes after a bullet is the entire proof for a subgoal. In this example, each of the subgoals is easily proved by a single use of reflexivity, which itself performs some simplification — e.g., the first one simplifies beq_nat (S n' + 1) 0 to false by first rewriting (S n' + 1) to S (n' + 1), then unfolding beq_nat, and then simplifying the match. Marking cases with bullets is entirely optional: if bullets are not present, Coq simply asks you to prove each subgoal in sequence, one at a time. But it is a good idea to use bullets. For one thing, they make the structure of a proof apparent, making it more readable. Also, bullets instruct Coq to ensure that a subgoal is complete before trying to verify the next one, preventing proofs for different subgoals from getting mixed up. These issues become especially important in large developments, where fragile proofs lead to long debugging sessions. There are no hard and fast rules for how proofs should be formatted in Coq — in particular, where lines should be broken and how sections of the proof should be indented to indicate their nested structure. However, if the places where multiple subgoals are generated are marked with explicit bullets at the beginning of lines, then the proof will be readable almost no matter what choices are made about other aspects of layout. This is also a good place to mention one other piece of somewhat obvious advice about line lengths. Beginning Coq users sometimes tend to the extremes, either writing each tactic on its own line or writing entire proofs on one line. Good style lies somewhere in the middle. One reasonable convention is to limit yourself to 80-character lines. The destruct tactic can be used with any inductively defined datatype. For example, we use it next to prove that boolean negation is involutive — i.e., that negation is its own inverse.

Note that the destruct here has no as clause because none of the subcases of the destruct need to bind any variables, so there is no need to specify any names. (We could also have written as [|], or as [].) In fact, we can omit the as clause from any destruct and Coq will fill in variable names automatically. This is generally considered bad style, since Coq often makes confusing choices of names when left to its own devices. It is sometimes useful to invoke destruct inside a subgoal, generating yet more proof obligations. In this case, we use different kinds of bullets to mark goals on different "levels." For example:

Each pair of calls to reflexivity corresponds to the subgoals that were generated after the execution of the destruct c line right above it. Besides - and +, Coq proofs can also use * (asterisk) as a third kind of bullet. If we ever encounter a proof that generates more than three levels of subgoals, we can also enclose individual subgoals in curly braces ({ ... }):

Since curly braces mark both the beginning and the end of a proof, they can be used for multiple subgoal levels, as this example shows. Furthermore, curly braces allow us to reuse the same bullet shapes at multiple levels in a proof:

Before closing the chapter, let's mention one final convenience. As you may have noticed, many proofs perform case analysis on a variable right after introducing it: This pattern is so common that Coq provides a shorthand for it: we can perform case analysis on a variable when introducing it by using an intro pattern instead of a variable name. For instance, here is a shorter proof of the plus_1_neq_0 theorem above.

If there are no arguments to name, we can just write [].

Exercise: 2 stars (andb_true_elim2)

Prove the following claim, marking cases (and subcases) with bullets when you use destruct.

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Exercise: 1 star (zero_nbeq_plus_1)

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More on Notation (Optional)

(In general, sections marked Optional are not needed to follow the rest of the book, except possibly other Optional sections. On a first reading, you might want to skim these sections so that you know what's there for future reference.) Recall the notation definitions for infix plus and times:

For each notation symbol in Coq, we can specify its precedence level and its associativity. The precedence level n is specified by writing at level n; this helps Coq parse compound expressions. The associativity setting helps to disambiguate expressions containing multiple occurrences of the same symbol. For example, the parameters specified above for + and * say that the expression 1+2*3*4 is shorthand for (1+((2*3)*4)). Coq uses precedence levels from 0 to 100, and left, right, or no associativity. We will see more examples of this later, e.g., in the Lists chapter. Each notation symbol is also associated with a notation scope. Coq tries to guess what scope is meant from context, so when it sees S(O*O) it guesses nat_scope, but when it sees the cartesian product (tuple) type bool*bool it guesses type_scope. Occasionally, it is necessary to help it out with percent-notation by writing (x*y)%nat, and sometimes in what Coq prints it will use %nat to indicate what scope a notation is in. Notation scopes also apply to numeral notation (3, 4, 5, etc.), so you may sometimes see 0%nat, which means O (the natural number 0 that we're using in this chapter), or 0%Z, which means the Integer zero (which comes from a different part of the standard library).

Fixpoints and Structural Recursion (Optional)

Here is a copy of the definition of addition:

When Coq checks this definition, it notes that plus' is "decreasing on 1st argument." What this means is that we are performing a structural recursion over the argument n — i.e., that we make recursive calls only on strictly smaller values of n. This implies that all calls to plus' will eventually terminate. Coq demands that some argument of every Fixpoint definition is "decreasing." This requirement is a fundamental feature of Coq's design: In particular, it guarantees that every function that can be defined in Coq will terminate on all inputs. However, because Coq's "decreasing analysis" is not very sophisticated, it is sometimes necessary to write functions in slightly unnatural ways.

Exercise: 2 stars, optional (decreasing)

To get a concrete sense of this, find a way to write a sensible Fixpoint definition (of a simple function on numbers, say) that does terminate on all inputs, but that Coq will reject because of this restriction.

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More Exercises

Exercise: 2 stars (boolean_functions)

Use the tactics you have learned so far to prove the following theorem about boolean functions.

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Exercise: 3 stars (binary)

Consider a different, more efficient representation of natural numbers using a binary rather than unary system. That is, instead of saying that each natural number is either zero or the successor of a natural number, we can say that each binary number is either

• zero,
• twice a binary number, or
• one more than twice a binary number.

(a) First, write an inductive definition of the type bin corresponding to this description of binary numbers. (Hint: Recall that the definition of nat from class, says nothing about what O and S "mean." It just says "O is in the set called nat, and if n is in the set then so is S n." The interpretation of O as zero and S as successor/plus one comes from the way that we use nat values, by writing functions to do things with them, proving things about them, and so on. Your definition of bin should be correspondingly simple; it is the functions you will write next that will give it mathematical meaning.) (b) Next, write an increment function incr for binary numbers, and a function bin_to_nat to convert binary numbers to unary numbers. (c) Write five unit tests test_bin_incr1, test_bin_incr2, etc. for your increment and binary-to-unary functions. Notice that incrementing a binary number and then converting it to unary should yield the same result as first converting it to unary and then incrementing.