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author Adam Chlipala <adam@chlipala.net>
date Mon, 22 Oct 2012 14:23:52 -0400
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17 We would all like to have programs check that our programs are correct. Due in no small part to some bold but unfulfilled promises in the history of computer science, today most people who write software, practitioners and academics alike, assume that the costs of formal program verification outweigh the benefits. The purpose of this book is to convince you that the technology of program verification is mature enough today that it makes sense to use it in a support role in many kinds of research projects in computer science. Beyond the convincing, I also want to provide a handbook on practical engineering of certified programs with the Coq proof assistant. Almost every subject covered is also relevant to interactive computer theorem-proving in general, such as for traditional mathematical theorems. In fact, I hope to demonstrate how verified programs are useful as building blocks in all sorts of formalizations. 17 We would all like to have programs check that our programs are correct. Due in no small part to some bold but unfulfilled promises in the history of computer science, today most people who write software, practitioners and academics alike, assume that the costs of formal program verification outweigh the benefits. The purpose of this book is to convince you that the technology of program verification is mature enough today that it makes sense to use it in a support role in many kinds of research projects in computer science. Beyond the convincing, I also want to provide a handbook on practical engineering of certified programs with the Coq proof assistant. Almost every subject covered is also relevant to interactive computer theorem-proving in general, such as for traditional mathematical theorems. In fact, I hope to demonstrate how verified programs are useful as building blocks in all sorts of formalizations.
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19 Research into mechanized theorem proving began in the second half of the 20th century, and some of the earliest practical work involved Nqthm%~\cite{Nqthm}\index{Nqthm}%, the "Boyer-Moore Theorem Prover," which was used to prove such theorems as correctness of a complete hardware and software stack%~\cite{Piton}%. ACL2%~\cite{CAR}\index{ACL2}%, Nqthm's successor, has seen significant industry adoption, for instance, by AMD to verify correctness of floating-point division units%~\cite{AMD}%. 19 Research into mechanized theorem proving began in the second half of the 20th century, and some of the earliest practical work involved Nqthm%~\cite{Nqthm}\index{Nqthm}%, the "Boyer-Moore Theorem Prover," which was used to prove such theorems as correctness of a complete hardware and software stack%~\cite{Piton}%. ACL2%~\cite{CAR}\index{ACL2}%, Nqthm's successor, has seen significant industry adoption, for instance, by AMD to verify correctness of floating-point division units%~\cite{AMD}%.
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21 Around the beginning of the 21st century, the pace of progress in practical applications of interactive theorem proving accelerated significantly. Several well-known formal developments have been carried out in Coq, the system that this book deals with. In the realm of pure mathematics, Georges Gonthier built a machine-checked proof of the four color theorem%~\cite{4C}%, a mathematical problem first posed more than a hundred years before, where the only previous proofs had required trusting ad-hoc software to do brute-force checking of key facts. In the realm of program verification, Xavier Leroy led the CompCert project to produce a verified C compiler back-end%~\cite{CompCert}% robust enough to use with real embedded software. 21 Around the beginning of the 21st century, the pace of progress in practical applications of interactive theorem proving accelerated significantly. Several well-known formal developments have been carried out in Coq, the system that this book deals with. In the realm of pure mathematics, Georges Gonthier built a machine-checked proof of the four-color theorem%~\cite{4C}%, a mathematical problem first posed more than a hundred years before, where the only previous proofs had required trusting ad-hoc software to do brute-force checking of key facts. In the realm of program verification, Xavier Leroy led the CompCert project to produce a verified C compiler back-end%~\cite{CompCert}% robust enough to use with real embedded software.
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23 Many other recent projects have attracted attention by proving important theorems using computer proof assistant software. For instance, the L4.verified project%~\cite{seL4}% led by Gerwin Klein has given a mechanized proof of correctness for a realistic microkernel, using the Isabelle/HOL proof assistant%~\cite{Isabelle/HOL}\index{Isabelle/HOL}%. The amount of ongoing work in the area is so large that I cannot hope to list all the recent successes, so from this point I will assume that the reader is convinced both that we ought to want machine-checked proofs and that they seem to be feasible to produce. (To readers not yet convinced, I suggest a Web search for "machine-checked proof"!) 23 Many other recent projects have attracted attention by proving important theorems using computer proof assistant software. For instance, the L4.verified project%~\cite{seL4}% led by Gerwin Klein has given a mechanized proof of correctness for a realistic microkernel, using the Isabelle/HOL proof assistant%~\cite{Isabelle/HOL}\index{Isabelle/HOL}%. The amount of ongoing work in the area is so large that I cannot hope to list all the recent successes, so from this point I will assume that the reader is convinced both that we ought to want machine-checked proofs and that they seem to be feasible to produce. (To readers not yet convinced, I suggest a Web search for "machine-checked proof"!)
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25 The idea of %\index{certified program}% _certified program_ features prominently in this book's title. Here the word "certified" does _not_ refer to governmental rules for how the reliability of engineered systems may be demonstrated to sufficiently high standards. Rather, this concept of certification, a standard one in the programming languages and formal methods communities, has to do with the idea of a _certificate_, or formal mathematical artifact proving that a program meets its specification. Government certification procedures rarely provide strong mathematical guarantees, while certified programming provides guarantees about as strong as anything we could hope for. We trust the definition of a foundational mathematical logic, we trust an implementation of the logic, and we trust that we have encoded our informal intent properly in formal specifications, but little else is left open as an opportunity to certify incorrect software. For programs like compilers that run in batch mode, the notion of a %\index{certifying program}% _certifying_ program is also common, where each run of the program outputs both an answer and a proof that the answer is correct. Certified software can be considered to subsume certifying software, and this book focuses on the certified case, while also introducing principles and techniques of general interest for stating and proving theorems in Coq. 25 The idea of %\index{certified program}% _certified program_ features prominently in this book's title. Here the word "certified" does _not_ refer to governmental rules for how the reliability of engineered systems may be demonstrated to sufficiently high standards. Rather, this concept of certification, a standard one in the programming languages and formal methods communities, has to do with the idea of a _certificate_, or formal mathematical artifact proving that a program meets its specification. Government certification procedures rarely provide strong mathematical guarantees, while certified programming provides guarantees about as strong as anything we could hope for. We trust the definition of a foundational mathematical logic, we trust an implementation of that logic, and we trust that we have encoded our informal intent properly in formal specifications, but few other opportunities remain to certify incorrect software. For compilers and other programs that run in batch mode, the notion of a %\index{certifying program}% _certifying_ program is also common, where each run of the program outputs both an answer and a proof that the answer is correct. Any certifying program can be composed with a proof checker to produce a certified program, and this book focuses on the certified case, while also introducing principles and techniques of general interest for stating and proving theorems in Coq.
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27 %\medskip% 27 %\medskip%
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29 There are a good number of (though definitely not "many") tools that are in wide use today for building machine-checked mathematical proofs and machine-certified programs. The following is my attempt at an exhaustive list of interactive "proof assistants" satisfying a few criteria. First, the authors of each tool must intend for it to be put to use for software-related applications. Second, there must have been enough engineering effort put into the tool that someone not doing research on the tool itself would feel his time was well spent using it. A third criterion is more of an empirical validation of the second: the tool must have a significant user community outside of its own development team. 29 There are a good number of (though definitely not "many") tools that are in wide use today for building machine-checked mathematical proofs and machine-certified programs. The following is my attempt at an exhaustive list of interactive "proof assistants" satisfying a few criteria. First, the authors of each tool must intend for it to be put to use for software-related applications. Second, there must have been enough engineering effort put into the tool that someone not doing research on the tool itself would feel his time was well spent using it. A third criterion is more of an empirical validation of the second: the tool must have a significant user community outside of its own development team.
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58 58
59 (** * Why Coq? *) 59 (** * Why Coq? *)
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61 (** 61 (**
62 This book is going to be about certified programming using Coq, and I am convinced that it is the best tool for the job. Coq has a number of very attractive properties, which I will summarize here, mentioning which of the other candidate tools lack each property. 62 This book is going to be about certified programming using Coq, and I am convinced that it is the best tool for the job. Coq has a number of very attractive properties, which I will summarize here, mentioning which of the other candidate tools lack which properties.
63 *) 63 *)
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65 65
66 (** ** Based on a Higher-Order Functional Programming Language *) 66 (** ** Based on a Higher-Order Functional Programming Language *)
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68 (** 68 (**
69 %\index{higher-order vs. first-order languages}%There is no reason to give up the familiar comforts of functional programming when you start writing certified programs. All of the tools I listed are based on functional programming languages, which means you can use them without their proof-related aspects to write and run regular programs. 69 %\index{higher-order vs. first-order languages}%There is no reason to give up the familiar comforts of functional programming when you start writing certified programs. All of the tools I listed are based on functional programming languages, which means you can use them without their proof-related features to write and run regular programs.
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71 %\index{ACL2}%ACL2 is notable in this field for having only a _first-order_ language at its foundation. That is, you cannot work with functions over functions and all those other treats of functional programming. By giving up this facility, ACL2 can make broader assumptions about how well its proof automation will work, but we can generally recover the same advantages in other proof assistants when we happen to be programming in first-order fragments. 71 %\index{ACL2}%ACL2 is notable in this field for having only a _first-order_ language at its foundation. That is, you cannot work with functions over functions and all those other treats of functional programming. By giving up this facility, ACL2 can make broader assumptions about how well its proof automation will work, but we can generally recover the same advantages in other proof assistants when we happen to be programming in first-order fragments.
72 *) 72 *)
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75 (** ** Dependent Types *) 75 (** ** Dependent Types *)
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77 (** 77 (**
78 A language with _dependent types_ may include references to programs inside of types. For instance, the type of an array might include a program expression giving the size of the array, making it possible to verify absence of out-of-bounds accesses statically. Dependent types can go even further than this, effectively capturing any correctness property in a type. For instance, later in this book, we will see how to give a Mini-ML compiler a type that guarantees that it maps well-typed source programs to well-typed target programs. 78 A language with _dependent types_ may include references to programs inside of types. For instance, the type of an array might include a program expression giving the size of the array, making it possible to verify absence of out-of-bounds accesses statically. Dependent types can go even further than this, effectively capturing any correctness property in a type. For instance, later in this book, we will see how to give a Mini-ML compiler a type that guarantees that it maps well-typed source programs to well-typed target programs.
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80 %\index{ACL2}%ACL2 and %\index{HOL}%HOL lack dependent types outright. %\index{PVS}%PVS and %\index{Twelf}%Twelf each supports a different strict subset of Coq's dependent type language. Twelf's type language is restricted to a bare-bones, monomorphic lambda calculus, which places serious restrictions on how complicated _computations inside types_ can be. This restriction is important for the soundness argument behind Twelf's approach to representing and checking proofs. 80 %\index{ACL2}%ACL2 and %\index{HOL}%HOL lack dependent types outright. Each of %\index{PVS}%PVS and %\index{Twelf}%Twelf supports a different strict subset of Coq's dependent type language. Twelf's type language is restricted to a bare-bones, monomorphic lambda calculus, which places serious restrictions on how complicated _computations inside types_ can be. This restriction is important for the soundness argument behind Twelf's approach to representing and checking proofs.
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82 In contrast, %\index{PVS}%PVS's dependent types are much more general, but they are squeezed inside the single mechanism of _subset types_, where a normal type is refined by attaching a predicate over its elements. Each member of the subset type is an element of the base type that satisfies the predicate. Chapter 6 of this book introduces that style of programming in Coq, while the remaining chapters of Part II deal with features of dependent typing in Coq that go beyond what PVS supports. 82 In contrast, %\index{PVS}%PVS's dependent types are much more general, but they are squeezed inside the single mechanism of _subset types_, where a normal type is refined by attaching a predicate over its elements. Each member of the subset type is an element of the base type that satisfies the predicate. Chapter 6 of this book introduces that style of programming in Coq, while the remaining chapters of Part II deal with features of dependent typing in Coq that go beyond what PVS supports.
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84 Dependent types are not just useful because they help you express correctness properties in types. Dependent types also often let you write certified programs _without writing anything that looks like a proof_. Even with subset types, which for many contexts can be used to express any relevant property with enough acrobatics, the human driving the proof assistant usually has to build some proofs explicitly. Writing formal proofs is hard, so we want to avoid it as far as possible, so dependent types are invaluable. 84 Dependent types are useful not only because they help you express correctness properties in types. Dependent types also often let you write certified programs _without writing anything that looks like a proof_. Even with subset types, which for many contexts can be used to express any relevant property with enough acrobatics, the human driving the proof assistant usually has to build some proofs explicitly. Writing formal proofs is hard, so we want to avoid it as far as possible. Dependent types are invaluable for this purpose.
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86 *) 86 *)
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88 (** ** An Easy-to-Check Kernel Proof Language *) 88 (** ** An Easy-to-Check Kernel Proof Language *)
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