### annotate src/MoreDep.v @ 499:2d7ce9e011f4

Pass through Chapter 8
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adamc@83 10 (* begin hide *)
adamc@85 11 Require Import Arith Bool List.
adam@314 13 Require Import CpdtTactics MoreSpecif.
adamc@83 16 (* end hide *)
adamc@83 19 (** %\chapter{More Dependent Types}% *)
adam@425 21 (** Subset types and their relatives help us integrate verification with programming. Though they reorganize the certified programmer's workflow, they tend not to have deep effects on proofs. We write largely the same proofs as we would for classical verification, with some of the structure moved into the programs themselves. It turns out that, when we use dependent types to their full potential, we warp the development and proving process even more than that, picking up "free theorems" to the extent that often a certified program is hardly more complex than its uncertified counterpart in Haskell or ML.
adam@476 23 In particular, we have only scratched the tip of the iceberg that is Coq's inductive definition mechanism. The inductive types we have seen so far have their counterparts in the other proof assistants that we surveyed in Chapter 1. This chapter explores the strange new world of dependent inductive datatypes outside [Prop], a possibility that sets Coq apart from all of the competition not based on type theory. *)
adamc@84 26 (** * Length-Indexed Lists *)
adam@338 28 (** Many introductions to dependent types start out by showing how to use them to eliminate array bounds checks%\index{array bounds checks}%. When the type of an array tells you how many elements it has, your compiler can detect out-of-bounds dereferences statically. Since we are working in a pure functional language, the next best thing is length-indexed lists%\index{length-indexed lists}%, which the following code defines. *)
adamc@84 31 Variable A : Set.
adamc@84 33 Inductive ilist : nat -> Set :=
adamc@84 34 | Nil : ilist O
adamc@84 35 | Cons : forall n, A -> ilist n -> ilist (S n).
adamc@84 37 (** We see that, within its section, [ilist] is given type [nat -> Set]. Previously, every inductive type we have seen has either had plain [Set] as its type or has been a predicate with some type ending in [Prop]. The full generality of inductive definitions lets us integrate the expressivity of predicates directly into our normal programming.
adam@405 39 The [nat] argument to [ilist] tells us the length of the list. The types of [ilist]'s constructors tell us that a [Nil] list has length [O] and that a [Cons] list has length one greater than the length of its tail. We may apply [ilist] to any natural number, even natural numbers that are only known at runtime. It is this breaking of the%\index{phase distinction}% _phase distinction_ that characterizes [ilist] as _dependently typed_.
adamc@213 41 In expositions of list types, we usually see the length function defined first, but here that would not be a very productive function to code. Instead, let us implement list concatenation. *)
adamc@213 43 Fixpoint app n1 (ls1 : ilist n1) n2 (ls2 : ilist n2) : ilist (n1 + n2) :=
adamc@213 45 | Nil => ls2
adamc@213 46 | Cons _ x ls1' => Cons x (app ls1' ls2)
adam@338 49 (** Past Coq versions signalled an error for this definition. The code is still invalid within Coq's core language, but current Coq versions automatically add annotations to the original program, producing a valid core program. These are the annotations on [match] discriminees that we began to study in the previous chapter. We can rewrite [app] to give the annotations explicitly. *)
adamc@100 51 (* begin thide *)
adam@338 52 Fixpoint app' n1 (ls1 : ilist n1) n2 (ls2 : ilist n2) : ilist (n1 + n2) :=
adam@338 53 match ls1 in (ilist n1) return (ilist (n1 + n2)) with
adam@338 54 | Nil => ls2
adam@338 55 | Cons _ x ls1' => Cons x (app' ls1' ls2)
adamc@100 57 (* end thide *)
adam@398 59 (** Using [return] alone allowed us to express a dependency of the [match] result type on the _value_ of the discriminee. What %\index{Gallina terms!in}%[in] adds to our arsenal is a way of expressing a dependency on the _type_ of the discriminee. Specifically, the [n1] in the [in] clause above is a _binding occurrence_ whose scope is the [return] clause.
adam@398 61 We may use [in] clauses only to bind names for the arguments of an inductive type family. That is, each [in] clause must be an inductive type family name applied to a sequence of underscores and variable names of the proper length. The positions for _parameters_ to the type family must all be underscores. Parameters are those arguments declared with section variables or with entries to the left of the first colon in an inductive definition. They cannot vary depending on which constructor was used to build the discriminee, so Coq prohibits pointless matches on them. It is those arguments defined in the type to the right of the colon that we may name with [in] clauses.
adam@484 63 Our [app] function could be typed in so-called%\index{stratified type systems}% _stratified_ type systems, which avoid true dependency. That is, we could consider the length indices to lists to live in a separate, compile-time-only universe from the lists themselves. Compile-time data may be _erased_ such that we can still execute a program. As an example where erasure would not work, consider an injection function from regular lists to length-indexed lists. Here the run-time computation actually depends on details of the compile-time argument, if we decide that the list to inject can be considered compile-time. More commonly, we think of lists as run-time data. Neither case will work with %\%naive%{}% erasure. (It is not too important to grasp the details of this run-time/compile-time distinction, since Coq's expressive power comes from avoiding such restrictions.) *)
adamc@100 65 (* EX: Implement injection from normal lists *)
adamc@100 67 (* begin thide *)
adam@454 68 Fixpoint inject (ls : list A) : ilist (length ls) :=
adam@454 70 | nil => Nil
adam@454 71 | h :: t => Cons h (inject t)
adamc@84 74 (** We can define an inverse conversion and prove that it really is an inverse. *)
adam@454 76 Fixpoint unject n (ls : ilist n) : list A :=
adam@454 78 | Nil => nil
adam@454 79 | Cons _ h t => h :: unject t
adam@454 82 Theorem inject_inverse : forall ls, unject (inject ls) = ls.
adamc@100 85 (* end thide *)
adam@338 87 (* EX: Implement statically checked "car"/"hd" *)
adam@425 89 (** Now let us attempt a function that is surprisingly tricky to write. In ML, the list head function raises an exception when passed an empty list. With length-indexed lists, we can rule out such invalid calls statically, and here is a first attempt at doing so. We write [???] as a placeholder for a term that we do not know how to write, not for any real Coq notation like those introduced two chapters ago.
adam@454 91 Definition hd n (ls : ilist (S n)) : A :=
adam@454 93 | Nil => ???
adam@454 94 | Cons _ h _ => h
adamc@84 97 It is not clear what to write for the [Nil] case, so we are stuck before we even turn our function over to the type checker. We could try omitting the [Nil] case:
adam@454 99 Definition hd n (ls : ilist (S n)) : A :=
adam@454 101 | Cons _ h _ => h
adamc@84 106 Error: Non exhaustive pattern-matching: no clause found for pattern Nil
adam@480 109 Unlike in ML, we cannot use inexhaustive pattern matching, because there is no conception of a <<Match>> exception to be thrown. In fact, recent versions of Coq _do_ allow this, by implicit translation to a [match] that considers all constructors; the error message above was generated by an older Coq version. It is educational to discover for ourselves the encoding that the most recent Coq versions use. We might try using an [in] clause somehow.
adam@454 112 Definition hd n (ls : ilist (S n)) : A :=
adam@454 113 match ls in (ilist (S n)) with
adam@454 114 | Cons _ h _ => h
adam@398 122 In this and other cases, we feel like we want [in] clauses with type family arguments that are not variables. Unfortunately, Coq only supports variables in those positions. A completely general mechanism could only be supported with a solution to the problem of higher-order unification%~\cite{HOU}%, which is undecidable. There _are_ useful heuristics for handling non-variable indices which are gradually making their way into Coq, but we will spend some time in this and the next few chapters on effective pattern matching on dependent types using only the primitive [match] annotations.
adamc@84 124 Our final, working attempt at [hd] uses an auxiliary function and a surprising [return] annotation. *)
adamc@100 126 (* begin thide *)
adam@454 127 Definition hd' n (ls : ilist n) :=
adam@454 128 match ls in (ilist n) return (match n with O => unit | S _ => A end) with
adam@454 129 | Nil => tt
adam@454 130 | Cons _ h _ => h
adam@283 136 : forall n : nat, ilist n -> match n with
adam@283 137 | 0 => unit
adam@283 138 | S _ => A
adam@454 143 Definition hd n (ls : ilist (S n)) : A := hd' ls.
adamc@100 144 (* end thide *)
adamc@84 148 (** We annotate our main [match] with a type that is itself a [match]. We write that the function [hd'] returns [unit] when the list is empty and returns the carried type [A] in all other cases. In the definition of [hd], we just call [hd']. Because the index of [ls] is known to be nonzero, the type checker reduces the [match] in the type of [hd'] to [A]. *)
adam@371 151 (** * The One Rule of Dependent Pattern Matching in Coq *)
adam@405 153 (** The rest of this chapter will demonstrate a few other elegant applications of dependent types in Coq. Readers encountering such ideas for the first time often feel overwhelmed, concluding that there is some magic at work whereby Coq sometimes solves the halting problem for the programmer and sometimes does not, applying automated program understanding in a way far beyond what is found in conventional languages. The point of this section is to cut off that sort of thinking right now! Dependent type-checking in Coq follows just a few algorithmic rules. Chapters 10 and 12 introduce many of those rules more formally, and the main additional rule is centered on%\index{dependent pattern matching}% _dependent pattern matching_ of the kind we met in the previous section.
adam@405 155 A dependent pattern match is a [match] expression where the type of the overall [match] is a function of the value and/or the type of the%\index{discriminee}% _discriminee_, the value being matched on. In other words, the [match] type _depends_ on the discriminee.
adam@398 157 When exactly will Coq accept a dependent pattern match as well-typed? Some other dependently typed languages employ fancy decision procedures to determine when programs satisfy their very expressive types. The situation in Coq is just the opposite. Only very straightforward symbolic rules are applied. Such a design choice has its drawbacks, as it forces programmers to do more work to convince the type checker of program validity. However, the great advantage of a simple type checking algorithm is that its action on _invalid_ programs is easier to understand!
adam@371 159 We come now to the one rule of dependent pattern matching in Coq. A general dependent pattern match assumes this form (with unnecessary parentheses included to make the syntax easier to parse):
adam@480 161 match E as y in (T x1 ... xn) return U with
adam@371 162 | C z1 ... zm => B
adam@480 167 The discriminee is a term [E], a value in some inductive type family [T], which takes [n] arguments. An %\index{as clause}%[as] clause binds the name [y] to refer to the discriminee [E]. An %\index{in clause}%[in] clause binds an explicit name [xi] for the [i]th argument passed to [T] in the type of [E].
adam@480 169 We bind these new variables [y] and [xi] so that they may be referred to in [U], a type given in the %\index{return clause}%[return] clause. The overall type of the [match] will be [U], with [E] substituted for [y], and with each [xi] substituted by the actual argument appearing in that position within [E]'s type.
adam@371 171 In general, each case of a [match] may have a pattern built up in several layers from the constructors of various inductive type families. To keep this exposition simple, we will focus on patterns that are just single applications of inductive type constructors to lists of variables. Coq actually compiles the more general kind of pattern matching into this more restricted kind automatically, so understanding the typing of [match] requires understanding the typing of [match]es lowered to match one constructor at a time.
adam@371 173 The last piece of the typing rule tells how to type-check a [match] case. A generic constructor application [C z1 ... zm] has some type [T x1' ... xn'], an application of the type family used in [E]'s type, probably with occurrences of the [zi] variables. From here, a simple recipe determines what type we will require for the case body [B]. The type of [B] should be [U] with the following two substitutions applied: we replace [y] (the [as] clause variable) with [C z1 ... zm], and we replace each [xi] (the [in] clause variables) with [xi']. In other words, we specialize the result type based on what we learn based on which pattern has matched the discriminee.
adam@371 175 This is an exhaustive description of the ways to specify how to take advantage of which pattern has matched! No other mechanisms come into play. For instance, there is no way to specify that the types of certain free variables should be refined based on which pattern has matched. In the rest of the book, we will learn design patterns for achieving similar effects, where each technique leads to an encoding only in terms of [in], [as], and [return] clauses.
adam@425 177 A few details have been omitted above. In Chapter 3, we learned that inductive type families may have both%\index{parameters}% _parameters_ and regular arguments. Within an [in] clause, a parameter position must have the wildcard [_] written, instead of a variable. (In general, Coq uses wildcard [_]'s either to indicate pattern variables that will not be mentioned again or to indicate positions where we would like type inference to infer the appropriate terms.) Furthermore, recent Coq versions are adding more and more heuristics to infer dependent [match] annotations in certain conditions. The general annotation inference problem is undecidable, so there will always be serious limitations on how much work these heuristics can do. When in doubt about why a particular dependent [match] is failing to type-check, add an explicit [return] annotation! At that point, the mechanical rule sketched in this section will provide a complete account of "what the type checker is thinking." Be sure to avoid the common pitfall of writing a [return] annotation that does not mention any variables bound by [in] or [as]; such a [match] will never refine typing requirements based on which pattern has matched. (One simple exception to this rule is that, when the discriminee is a variable, that same variable may be treated as if it were repeated as an [as] clause.) *)
adamc@85 180 (** * A Tagless Interpreter *)
adam@405 182 (** A favorite example for motivating the power of functional programming is implementation of a simple expression language interpreter. In ML and Haskell, such interpreters are often implemented using an algebraic datatype of values, where at many points it is checked that a value was built with the right constructor of the value type. With dependent types, we can implement a%\index{tagless interpreters}% _tagless_ interpreter that both removes this source of runtime inefficiency and gives us more confidence that our implementation is correct. *)
adamc@85 184 Inductive type : Set :=
adamc@85 185 | Nat : type
adamc@85 186 | Bool : type
adamc@85 187 | Prod : type -> type -> type.
adamc@85 189 Inductive exp : type -> Set :=
adamc@85 190 | NConst : nat -> exp Nat
adamc@85 191 | Plus : exp Nat -> exp Nat -> exp Nat
adamc@85 192 | Eq : exp Nat -> exp Nat -> exp Bool
adamc@85 194 | BConst : bool -> exp Bool
adamc@85 195 | And : exp Bool -> exp Bool -> exp Bool
adamc@85 196 | If : forall t, exp Bool -> exp t -> exp t -> exp t
adamc@85 198 | Pair : forall t1 t2, exp t1 -> exp t2 -> exp (Prod t1 t2)
adamc@85 199 | Fst : forall t1 t2, exp (Prod t1 t2) -> exp t1
adamc@85 200 | Snd : forall t1 t2, exp (Prod t1 t2) -> exp t2.
adam@448 202 (** We have a standard algebraic datatype [type], defining a type language of naturals, Booleans, and product (pair) types. Then we have the indexed inductive type [exp], where the argument to [exp] tells us the encoded type of an expression. In effect, we are defining the typing rules for expressions simultaneously with the syntax.
adam@398 204 We can give types and expressions semantics in a new style, based critically on the chance for _type-level computation_. *)
adamc@85 206 Fixpoint typeDenote (t : type) : Set :=
adamc@85 208 | Nat => nat
adamc@85 209 | Bool => bool
adamc@85 210 | Prod t1 t2 => typeDenote t1 * typeDenote t2
adam@465 213 (** The [typeDenote] function compiles types of our object language into "native" Coq types. It is deceptively easy to implement. The only new thing we see is the [%]%\coqdocvar{%#<tt>#type#</tt>#%}% annotation, which tells Coq to parse the [match] expression using the notations associated with types. Without this annotation, the [*] would be interpreted as multiplication on naturals, rather than as the product type constructor. The token %\coqdocvar{%#<tt>#type#</tt>#%}% is one example of an identifier bound to a%\index{notation scope delimiter}% _notation scope delimiter_. In this book, we will not go into more detail on notation scopes, but the Coq manual can be consulted for more information.
adamc@85 215 We can define a function [expDenote] that is typed in terms of [typeDenote]. *)
adamc@213 217 Fixpoint expDenote t (e : exp t) : typeDenote t :=
adamc@85 219 | NConst n => n
adamc@85 220 | Plus e1 e2 => expDenote e1 + expDenote e2
adamc@85 221 | Eq e1 e2 => if eq_nat_dec (expDenote e1) (expDenote e2) then true else false
adamc@85 223 | BConst b => b
adamc@85 224 | And e1 e2 => expDenote e1 && expDenote e2
adamc@85 225 | If _ e' e1 e2 => if expDenote e' then expDenote e1 else expDenote e2
adamc@85 227 | Pair _ _ e1 e2 => (expDenote e1, expDenote e2)
adamc@85 228 | Fst _ _ e' => fst (expDenote e')
adamc@85 229 | Snd _ _ e' => snd (expDenote e')
adam@437 232 (* begin hide *)
adam@437 233 (* begin thide *)
adam@437 234 Definition sumboool := sumbool.
adam@437 235 (* end thide *)
adam@437 236 (* end hide *)
adam@448 238 (** Despite the fancy type, the function definition is routine. In fact, it is less complicated than what we would write in ML or Haskell 98, since we do not need to worry about pushing final values in and out of an algebraic datatype. The only unusual thing is the use of an expression of the form [if E then true else false] in the [Eq] case. Remember that [eq_nat_dec] has a rich dependent type, rather than a simple Boolean type. Coq's native [if] is overloaded to work on a test of any two-constructor type, so we can use [if] to build a simple Boolean from the [sumbool] that [eq_nat_dec] returns.
adamc@85 240 We can implement our old favorite, a constant folding function, and prove it correct. It will be useful to write a function [pairOut] that checks if an [exp] of [Prod] type is a pair, returning its two components if so. Unsurprisingly, a first attempt leads to a type error.
adamc@85 242 Definition pairOut t1 t2 (e : exp (Prod t1 t2)) : option (exp t1 * exp t2) :=
adamc@85 243 match e in (exp (Prod t1 t2)) return option (exp t1 * exp t2) with
adamc@85 244 | Pair _ _ e1 e2 => Some (e1, e2)
adamc@85 245 | _ => None
adamc@85 253 We run again into the problem of not being able to specify non-variable arguments in [in] clauses. The problem would just be hopeless without a use of an [in] clause, though, since the result type of the [match] depends on an argument to [exp]. Our solution will be to use a more general type, as we did for [hd]. First, we define a type-valued function to use in assigning a type to [pairOut]. *)
adamc@100 255 (* EX: Define a function [pairOut : forall t1 t2, exp (Prod t1 t2) -> option (exp t1 * exp t2)] *)
adamc@100 257 (* begin thide *)
adam@499 258 Definition pairOutType (t : type) := option (match t with
adam@499 259 | Prod t1 t2 => exp t1 * exp t2
adam@499 260 | _ => unit
adam@499 263 (** When passed a type that is a product, [pairOutType] returns our final desired type. On any other input type, [pairOutType] returns the harmless [option unit], since we do not care about extracting components of non-pairs. Now [pairOut] is easy to write. *)
adamc@85 265 Definition pairOut t (e : exp t) :=
adamc@85 266 match e in (exp t) return (pairOutType t) with
adamc@85 267 | Pair _ _ e1 e2 => Some (e1, e2)
adam@499 268 | _ => None
adamc@100 270 (* end thide *)
adam@499 272 (** With [pairOut] available, we can write [cfold] in a straightforward way. There are really no surprises beyond that Coq verifies that this code has such an expressive type, given the small annotation burden. In some places, we see that Coq's [match] annotation inference is too smart for its own good, and we have to turn that inference off with explicit [return] clauses. *)
adamc@204 274 Fixpoint cfold t (e : exp t) : exp t :=
adamc@85 276 | NConst n => NConst n
adamc@85 277 | Plus e1 e2 =>
adamc@85 278 let e1' := cfold e1 in
adamc@85 279 let e2' := cfold e2 in
adam@417 280 match e1', e2' return exp Nat with
adamc@85 281 | NConst n1, NConst n2 => NConst (n1 + n2)
adamc@85 282 | _, _ => Plus e1' e2'
adamc@85 284 | Eq e1 e2 =>
adamc@85 285 let e1' := cfold e1 in
adamc@85 286 let e2' := cfold e2 in
adam@417 287 match e1', e2' return exp Bool with
adamc@85 288 | NConst n1, NConst n2 => BConst (if eq_nat_dec n1 n2 then true else false)
adamc@85 289 | _, _ => Eq e1' e2'
adamc@85 292 | BConst b => BConst b
adamc@85 293 | And e1 e2 =>
adamc@85 294 let e1' := cfold e1 in
adamc@85 295 let e2' := cfold e2 in
adam@417 296 match e1', e2' return exp Bool with
adamc@85 297 | BConst b1, BConst b2 => BConst (b1 && b2)
adamc@85 298 | _, _ => And e1' e2'
adamc@85 300 | If _ e e1 e2 =>
adamc@85 301 let e' := cfold e in
adamc@85 303 | BConst true => cfold e1
adamc@85 304 | BConst false => cfold e2
adamc@85 305 | _ => If e' (cfold e1) (cfold e2)
adamc@85 308 | Pair _ _ e1 e2 => Pair (cfold e1) (cfold e2)
adamc@85 309 | Fst _ _ e =>
adamc@85 310 let e' := cfold e in
adamc@85 311 match pairOut e' with
adamc@85 312 | Some p => fst p
adamc@85 313 | None => Fst e'
adamc@85 315 | Snd _ _ e =>
adamc@85 316 let e' := cfold e in
adamc@85 317 match pairOut e' with
adamc@85 318 | Some p => snd p
adamc@85 319 | None => Snd e'
adamc@85 323 (** The correctness theorem for [cfold] turns out to be easy to prove, once we get over one serious hurdle. *)
adamc@85 325 Theorem cfold_correct : forall t (e : exp t), expDenote e = expDenote (cfold e).
adamc@100 326 (* begin thide *)
adamc@85 329 (** The first remaining subgoal is:
adamc@85 332 expDenote (cfold e1) + expDenote (cfold e2) =
adamc@85 334 match cfold e1 with
adamc@85 335 | NConst n1 =>
adamc@85 336 match cfold e2 with
adamc@85 337 | NConst n2 => NConst (n1 + n2)
adamc@85 338 | Plus _ _ => Plus (cfold e1) (cfold e2)
adamc@85 339 | Eq _ _ => Plus (cfold e1) (cfold e2)
adamc@85 340 | BConst _ => Plus (cfold e1) (cfold e2)
adamc@85 341 | And _ _ => Plus (cfold e1) (cfold e2)
adamc@85 342 | If _ _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 343 | Pair _ _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 344 | Fst _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 345 | Snd _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 347 | Plus _ _ => Plus (cfold e1) (cfold e2)
adamc@85 348 | Eq _ _ => Plus (cfold e1) (cfold e2)
adamc@85 349 | BConst _ => Plus (cfold e1) (cfold e2)
adamc@85 350 | And _ _ => Plus (cfold e1) (cfold e2)
adamc@85 351 | If _ _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 352 | Pair _ _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 353 | Fst _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 354 | Snd _ _ _ => Plus (cfold e1) (cfold e2)
adam@454 359 We would like to do a case analysis on [cfold e1], and we attempt to do so in the way that has worked so far.
adamc@85 365 User error: e1 is used in hypothesis e
adamc@85 368 Coq gives us another cryptic error message. Like so many others, this one basically means that Coq is not able to build some proof about dependent types. It is hard to generate helpful and specific error messages for problems like this, since that would require some kind of understanding of the dependency structure of a piece of code. We will encounter many examples of case-specific tricks for recovering from errors like this one.
adam@499 370 For our current proof, we can use a tactic [dep_destruct]%\index{tactics!dep\_destruct}% defined in the book's [CpdtTactics] module. General elimination/inversion of dependently typed hypotheses is undecidable, as witnessed by a simple reduction from the known-undecidable problem of higher-order unification, which has come up a few times already. The tactic [dep_destruct] makes a best effort to handle some common cases, relying upon the more primitive %\index{tactics!dependent destruction}%[dependent destruction] tactic that comes with Coq. In a future chapter, we will learn about the explicit manipulation of equality proofs that is behind [dependent destruction]'s implementation, but for now, we treat it as a useful black box. (In Chapter 12, we will also see how [dependent destruction] forces us to make a larger philosophical commitment about our logic than we might like, and we will see some workarounds.) *)
adamc@85 374 (** This successfully breaks the subgoal into 5 new subgoals, one for each constructor of [exp] that could produce an [exp Nat]. Note that [dep_destruct] is successful in ruling out the other cases automatically, in effect automating some of the work that we have done manually in implementing functions like [hd] and [pairOut].
adam@480 376 This is the only new trick we need to learn to complete the proof. We can back up and give a short, automated proof (which again is safe to skip and uses Ltac features not introduced yet). *)
adamc@85 381 repeat (match goal with
adam@405 382 | [ |- context[match cfold ?E with NConst _ => _ | _ => _ end] ] =>
adamc@213 384 | [ |- context[match pairOut (cfold ?E) with Some _ => _
adamc@213 385 | None => _ end] ] =>
adamc@85 387 | [ |- (if ?E then _ else _) = _ ] => destruct E
adamc@100 390 (* end thide *)
adam@405 392 (** With this example, we get a first taste of how to build automated proofs that adapt automatically to changes in function definitions. *)
adam@338 395 (** * Dependently Typed Red-Black Trees *)
adam@475 397 (** Red-black trees are a favorite purely functional data structure with an interesting invariant. We can use dependent types to guarantee that operations on red-black trees preserve the invariant. For simplicity, we specialize our red-black trees to represent sets of [nat]s. *)
adamc@94 399 Inductive color : Set := Red | Black.
adamc@94 401 Inductive rbtree : color -> nat -> Set :=
adamc@94 402 | Leaf : rbtree Black 0
adamc@214 403 | RedNode : forall n, rbtree Black n -> nat -> rbtree Black n -> rbtree Red n
adamc@94 404 | BlackNode : forall c1 c2 n, rbtree c1 n -> nat -> rbtree c2 n -> rbtree Black (S n).
adam@476 406 (** A value of type [rbtree c d] is a red-black tree whose root has color [c] and that has black depth [d]. The latter property means that there are exactly [d] black-colored nodes on any path from the root to a leaf. *)
adamc@214 408 (** At first, it can be unclear that this choice of type indices tracks any useful property. To convince ourselves, we will prove that every red-black tree is balanced. We will phrase our theorem in terms of a depth calculating function that ignores the extra information in the types. It will be useful to parameterize this function over a combining operation, so that we can re-use the same code to calculate the minimum or maximum height among all paths from root to leaf. *)
adamc@100 410 (* EX: Prove that every [rbtree] is balanced. *)
adamc@100 412 (* begin thide *)
adamc@95 413 Require Import Max Min.
adamc@95 416 Variable f : nat -> nat -> nat.
adamc@214 418 Fixpoint depth c n (t : rbtree c n) : nat :=
adamc@95 420 | Leaf => 0
adamc@95 421 | RedNode _ t1 _ t2 => S (f (depth t1) (depth t2))
adamc@95 422 | BlackNode _ _ _ t1 _ t2 => S (f (depth t1) (depth t2))
adam@338 426 (** Our proof of balanced-ness decomposes naturally into a lower bound and an upper bound. We prove the lower bound first. Unsurprisingly, a tree's black depth provides such a bound on the minimum path length. We use the richly typed procedure [min_dec] to do case analysis on whether [min X Y] equals [X] or [Y]. *)
adam@283 431 : forall n m : nat, {min n m = n} + {min n m = m}
adamc@95 435 Theorem depth_min : forall c n (t : rbtree c n), depth min t >= n.
adamc@95 438 | [ |- context[min ?X ?Y] ] => destruct (min_dec X Y)
adamc@214 442 (** There is an analogous upper-bound theorem based on black depth. Unfortunately, a symmetric proof script does not suffice to establish it. *)
adamc@214 444 Theorem depth_max : forall c n (t : rbtree c n), depth max t <= 2 * n + 1.
adamc@214 447 | [ |- context[max ?X ?Y] ] => destruct (max_dec X Y)
adamc@214 450 (** Two subgoals remain. One of them is: [[
adamc@214 452 t1 : rbtree Black n
adamc@214 454 t2 : rbtree Black n
adamc@214 455 IHt1 : depth max t1 <= n + (n + 0) + 1
adamc@214 456 IHt2 : depth max t2 <= n + (n + 0) + 1
adamc@214 457 e : max (depth max t1) (depth max t2) = depth max t1
adamc@214 459 S (depth max t1) <= n + (n + 0) + 1
adam@398 463 We see that [IHt1] is _almost_ the fact we need, but it is not quite strong enough. We will need to strengthen our induction hypothesis to get the proof to go through. *)
adamc@214 467 (** In particular, we prove a lemma that provides a stronger upper bound for trees with black root nodes. We got stuck above in a case about a red root node. Since red nodes have only black children, our IH strengthening will enable us to finish the proof. *)
adamc@95 469 Lemma depth_max' : forall c n (t : rbtree c n), match c with
adamc@95 470 | Red => depth max t <= 2 * n + 1
adamc@95 471 | Black => depth max t <= 2 * n
adamc@95 475 | [ |- context[max ?X ?Y] ] => destruct (max_dec X Y)
adamc@100 477 repeat (match goal with
adamc@214 478 | [ H : context[match ?C with Red => _ | Black => _ end] |- _ ] =>
adam@338 483 (** The original theorem follows easily from the lemma. We use the tactic %\index{tactics!generalize}%[generalize pf], which, when [pf] proves the proposition [P], changes the goal from [Q] to [P -> Q]. This transformation is useful because it makes the truth of [P] manifest syntactically, so that automation machinery can rely on [P], even if that machinery is not smart enough to establish [P] on its own. *)
adamc@95 485 Theorem depth_max : forall c n (t : rbtree c n), depth max t <= 2 * n + 1.
adamc@95 486 intros; generalize (depth_max' t); destruct c; crush.
adamc@214 489 (** The final balance theorem establishes that the minimum and maximum path lengths of any tree are within a factor of two of each other. *)
adamc@95 491 Theorem balanced : forall c n (t : rbtree c n), 2 * depth min t + 1 >= depth max t.
adamc@95 492 intros; generalize (depth_min t); generalize (depth_max t); crush.
adamc@100 494 (* end thide *)
adamc@214 496 (** Now we are ready to implement an example operation on our trees, insertion. Insertion can be thought of as breaking the tree invariants locally but then rebalancing. In particular, in intermediate states we find red nodes that may have red children. The type [rtree] captures the idea of such a node, continuing to track black depth as a type index. *)
adamc@94 498 Inductive rtree : nat -> Set :=
adamc@94 499 | RedNode' : forall c1 c2 n, rbtree c1 n -> nat -> rbtree c2 n -> rtree n.
adam@338 501 (** Before starting to define [insert], we define predicates capturing when a data value is in the set represented by a normal or possibly invalid tree. *)
adamc@96 504 Variable x : nat.
adamc@214 506 Fixpoint present c n (t : rbtree c n) : Prop :=
adamc@96 508 | Leaf => False
adamc@96 509 | RedNode _ a y b => present a \/ x = y \/ present b
adamc@96 510 | BlackNode _ _ _ a y b => present a \/ x = y \/ present b
adamc@96 513 Definition rpresent n (t : rtree n) : Prop :=
adamc@96 515 | RedNode' _ _ _ a y b => present a \/ x = y \/ present b
adam@338 519 (** Insertion relies on two balancing operations. It will be useful to give types to these operations using a relative of the subset types from last chapter. While subset types let us pair a value with a proof about that value, here we want to pair a value with another non-proof dependently typed value. The %\index{Gallina terms!sigT}%[sigT] type fills this role. *)
adamc@100 521 Locate "{ _ : _ & _ }".
adamc@214 524 "{ x : A & P }" := sigT (fun x : A => P)
adamc@214 530 Inductive sigT (A : Type) (P : A -> Type) : Type :=
adamc@214 531 existT : forall x : A, P x -> sigT P
adamc@214 535 (** It will be helpful to define a concise notation for the constructor of [sigT]. *)
adamc@94 537 Notation "{< x >}" := (existT _ _ x).
adamc@214 539 (** Each balance function is used to construct a new tree whose keys include the keys of two input trees, as well as a new key. One of the two input trees may violate the red-black alternation invariant (that is, it has an [rtree] type), while the other tree is known to be valid. Crucially, the two input trees have the same black depth.
adam@338 541 A balance operation may return a tree whose root is of either color. Thus, we use a [sigT] type to package the result tree with the color of its root. Here is the definition of the first balance operation, which applies when the possibly invalid [rtree] belongs to the left of the valid [rbtree].
adam@425 543 A quick word of encouragement: After writing this code, even I do not understand the precise details of how balancing works! I consulted Chris Okasaki's paper "Red-Black Trees in a Functional Setting" %\cite{Okasaki} %and transcribed the code to use dependent types. Luckily, the details are not so important here; types alone will tell us that insertion preserves balanced-ness, and we will prove that insertion produces trees containing the right keys.*)
adamc@94 545 Definition balance1 n (a : rtree n) (data : nat) c2 :=
adamc@214 546 match a in rtree n return rbtree c2 n
adamc@214 547 -> { c : color & rbtree c (S n) } with
adam@380 548 | RedNode' _ c0 _ t1 y t2 =>
adam@380 549 match t1 in rbtree c n return rbtree c0 n -> rbtree c2 n
adamc@214 550 -> { c : color & rbtree c (S n) } with
adamc@214 551 | RedNode _ a x b => fun c d =>
adamc@214 552 {<RedNode (BlackNode a x b) y (BlackNode c data d)>}
adamc@94 553 | t1' => fun t2 =>
adam@380 554 match t2 in rbtree c n return rbtree Black n -> rbtree c2 n
adamc@214 555 -> { c : color & rbtree c (S n) } with
adamc@214 556 | RedNode _ b x c => fun a d =>
adamc@214 557 {<RedNode (BlackNode a y b) x (BlackNode c data d)>}
adamc@95 558 | b => fun a t => {<BlackNode (RedNode a y b) data t>}
adam@405 563 (** We apply a trick that I call the%\index{convoy pattern}% _convoy pattern_. Recall that [match] annotations only make it possible to describe a dependence of a [match] _result type_ on the discriminee. There is no automatic refinement of the types of free variables. However, it is possible to effect such a refinement by finding a way to encode free variable type dependencies in the [match] result type, so that a [return] clause can express the connection.
adam@425 565 In particular, we can extend the [match] to return _functions over the free variables whose types we want to refine_. In the case of [balance1], we only find ourselves wanting to refine the type of one tree variable at a time. We match on one subtree of a node, and we want the type of the other subtree to be refined based on what we learn. We indicate this with a [return] clause starting like [rbtree _ n -> ...], where [n] is bound in an [in] pattern. Such a [match] expression is applied immediately to the "old version" of the variable to be refined, and the type checker is happy.
adam@338 567 Here is the symmetric function [balance2], for cases where the possibly invalid tree appears on the right rather than on the left. *)
adamc@94 569 Definition balance2 n (a : rtree n) (data : nat) c2 :=
adamc@94 570 match a in rtree n return rbtree c2 n -> { c : color & rbtree c (S n) } with
adam@380 571 | RedNode' _ c0 _ t1 z t2 =>
adam@380 572 match t1 in rbtree c n return rbtree c0 n -> rbtree c2 n
adamc@214 573 -> { c : color & rbtree c (S n) } with
adamc@214 574 | RedNode _ b y c => fun d a =>
adamc@214 575 {<RedNode (BlackNode a data b) y (BlackNode c z d)>}
adamc@94 576 | t1' => fun t2 =>
adam@380 577 match t2 in rbtree c n return rbtree Black n -> rbtree c2 n
adamc@214 578 -> { c : color & rbtree c (S n) } with
adamc@214 579 | RedNode _ c z' d => fun b a =>
adamc@214 580 {<RedNode (BlackNode a data b) z (BlackNode c z' d)>}
adamc@95 581 | b => fun a t => {<BlackNode t data (RedNode a z b)>}
adamc@214 586 (** Now we are almost ready to get down to the business of writing an [insert] function. First, we enter a section that declares a variable [x], for the key we want to insert. *)
adamc@94 589 Variable x : nat.
adamc@214 591 (** Most of the work of insertion is done by a helper function [ins], whose return types are expressed using a type-level function [insResult]. *)
adamc@94 593 Definition insResult c n :=
adamc@94 595 | Red => rtree n
adamc@94 596 | Black => { c' : color & rbtree c' n }
adam@338 599 (** That is, inserting into a tree with root color [c] and black depth [n], the variety of tree we get out depends on [c]. If we started with a red root, then we get back a possibly invalid tree of depth [n]. If we started with a black root, we get back a valid tree of depth [n] with a root node of an arbitrary color.
adamc@214 601 Here is the definition of [ins]. Again, we do not want to dwell on the functional details. *)
adamc@214 603 Fixpoint ins c n (t : rbtree c n) : insResult c n :=
adamc@94 605 | Leaf => {< RedNode Leaf x Leaf >}
adamc@94 606 | RedNode _ a y b =>
adamc@94 607 if le_lt_dec x y
adamc@94 608 then RedNode' (projT2 (ins a)) y b
adamc@94 609 else RedNode' a y (projT2 (ins b))
adamc@94 610 | BlackNode c1 c2 _ a y b =>
adamc@94 611 if le_lt_dec x y
adamc@94 613 match c1 return insResult c1 _ -> _ with
adamc@94 614 | Red => fun ins_a => balance1 ins_a y b
adamc@94 615 | _ => fun ins_a => {< BlackNode (projT2 ins_a) y b >}
adamc@94 618 match c2 return insResult c2 _ -> _ with
adamc@94 619 | Red => fun ins_b => balance2 ins_b y a
adamc@94 620 | _ => fun ins_b => {< BlackNode a y (projT2 ins_b) >}
adam@479 624 (** The one new trick is a variation of the convoy pattern. In each of the last two pattern matches, we want to take advantage of the typing connection between the trees [a] and [b]. We might %\%naive%{}%ly apply the convoy pattern directly on [a] in the first [match] and on [b] in the second. This satisfies the type checker per se, but it does not satisfy the termination checker. Inside each [match], we would be calling [ins] recursively on a locally bound variable. The termination checker is not smart enough to trace the dataflow into that variable, so the checker does not know that this recursive argument is smaller than the original argument. We make this fact clearer by applying the convoy pattern on _the result of a recursive call_, rather than just on that call's argument.
adamc@214 626 Finally, we are in the home stretch of our effort to define [insert]. We just need a few more definitions of non-recursive functions. First, we need to give the final characterization of [insert]'s return type. Inserting into a red-rooted tree gives a black-rooted tree where black depth has increased, and inserting into a black-rooted tree gives a tree where black depth has stayed the same and where the root is an arbitrary color. *)
adamc@94 628 Definition insertResult c n :=
adamc@94 630 | Red => rbtree Black (S n)
adamc@94 631 | Black => { c' : color & rbtree c' n }
adamc@214 634 (** A simple clean-up procedure translates [insResult]s into [insertResult]s. *)
adamc@97 636 Definition makeRbtree c n : insResult c n -> insertResult c n :=
adamc@94 638 | Red => fun r =>
adamc@94 640 | RedNode' _ _ _ a x b => BlackNode a x b
adamc@94 642 | Black => fun r => r
adamc@214 645 (** We modify Coq's default choice of implicit arguments for [makeRbtree], so that we do not need to specify the [c] and [n] arguments explicitly in later calls. *)
adamc@97 647 Implicit Arguments makeRbtree [c n].
adamc@214 649 (** Finally, we define [insert] as a simple composition of [ins] and [makeRbtree]. *)
adamc@94 651 Definition insert c n (t : rbtree c n) : insertResult c n :=
adamc@214 654 (** As we noted earlier, the type of [insert] guarantees that it outputs balanced trees whose depths have not increased too much. We also want to know that [insert] operates correctly on trees interpreted as finite sets, so we finish this section with a proof of that fact. *)
adamc@95 657 Variable z : nat.
adamc@214 659 (** The variable [z] stands for an arbitrary key. We will reason about [z]'s presence in particular trees. As usual, outside the section the theorems we prove will quantify over all possible keys, giving us the facts we wanted.
adam@367 661 We start by proving the correctness of the balance operations. It is useful to define a custom tactic [present_balance] that encapsulates the reasoning common to the two proofs. We use the keyword %\index{Vernacular commands!Ltac}%[Ltac] to assign a name to a proof script. This particular script just iterates between [crush] and identification of a tree that is being pattern-matched on and should be destructed. *)
adamc@98 665 repeat (match goal with
adam@425 666 | [ _ : context[match ?T with Leaf => _ | _ => _ end] |- _ ] =>
adam@405 668 | [ |- context[match ?T with Leaf => _ | _ => _ end] ] => dep_destruct T
adamc@214 671 (** The balance correctness theorems are simple first-order logic equivalences, where we use the function [projT2] to project the payload of a [sigT] value. *)
adam@294 673 Lemma present_balance1 : forall n (a : rtree n) (y : nat) c2 (b : rbtree c2 n),
adamc@95 674 present z (projT2 (balance1 a y b))
adamc@95 675 <-> rpresent z a \/ z = y \/ present z b.
adamc@213 679 Lemma present_balance2 : forall n (a : rtree n) (y : nat) c2 (b : rbtree c2 n),
adamc@95 680 present z (projT2 (balance2 a y b))
adamc@95 681 <-> rpresent z a \/ z = y \/ present z b.
adamc@214 685 (** To state the theorem for [ins], it is useful to define a new type-level function, since [ins] returns different result types based on the type indices passed to it. Recall that [x] is the section variable standing for the key we are inserting. *)
adamc@95 687 Definition present_insResult c n :=
adamc@95 688 match c return (rbtree c n -> insResult c n -> Prop) with
adamc@95 689 | Red => fun t r => rpresent z r <-> z = x \/ present z t
adamc@95 690 | Black => fun t r => present z (projT2 r) <-> z = x \/ present z t
adamc@214 693 (** Now the statement and proof of the [ins] correctness theorem are straightforward, if verbose. We proceed by induction on the structure of a tree, followed by finding case analysis opportunities on expressions we see being analyzed in [if] or [match] expressions. After that, we pattern-match to find opportunities to use the theorems we proved about balancing. Finally, we identify two variables that are asserted by some hypothesis to be equal, and we use that hypothesis to replace one variable with the other everywhere. *)
adamc@95 695 Theorem present_ins : forall c n (t : rbtree c n),
adamc@95 696 present_insResult t (ins t).
adamc@95 698 repeat (match goal with
adam@338 699 | [ _ : context[if ?E then _ else _] |- _ ] => destruct E
adamc@95 700 | [ |- context[if ?E then _ else _] ] => destruct E
adam@338 701 | [ _ : context[match ?C with Red => _ | Black => _ end]
adamc@214 702 |- _ ] => destruct C
adamc@95 704 try match goal with
adam@338 705 | [ _ : context[balance1 ?A ?B ?C] |- _ ] =>
adamc@95 706 generalize (present_balance1 A B C)
adamc@95 708 try match goal with
adam@338 709 | [ _ : context[balance2 ?A ?B ?C] |- _ ] =>
adamc@95 710 generalize (present_balance2 A B C)
adamc@95 712 try match goal with
adamc@95 713 | [ |- context[balance1 ?A ?B ?C] ] =>
adamc@95 714 generalize (present_balance1 A B C)
adamc@95 716 try match goal with
adamc@95 717 | [ |- context[balance2 ?A ?B ?C] ] =>
adamc@95 718 generalize (present_balance2 A B C)
adamc@95 722 | [ z : nat, x : nat |- _ ] =>
adamc@95 724 | [ H : z = x |- _ ] => rewrite H in *; clear H
adamc@214 730 (** The hard work is done. The most readable way to state correctness of [insert] involves splitting the property into two color-specific theorems. We write a tactic to encapsulate the reasoning steps that work to establish both facts. *)
adamc@213 733 unfold insert; intros n t; inversion t;
adamc@97 734 generalize (present_ins t); simpl;
adamc@97 735 dep_destruct (ins t); tauto.
adamc@95 737 Theorem present_insert_Red : forall n (t : rbtree Red n),
adamc@95 738 present z (insert t)
adamc@95 739 <-> (z = x \/ present z t).
adamc@95 743 Theorem present_insert_Black : forall n (t : rbtree Black n),
adamc@95 744 present z (projT2 (insert t))
adamc@95 745 <-> (z = x \/ present z t).
adam@454 751 (** We can generate executable OCaml code with the command %\index{Vernacular commands!Recursive Extraction}%[Recursive Extraction insert], which also automatically outputs the OCaml versions of all of [insert]'s dependencies. In our previous extractions, we wound up with clean OCaml code. Here, we find uses of %\index{Obj.magic}%<<Obj.magic>>, OCaml's unsafe cast operator for tweaking the apparent type of an expression in an arbitrary way. Casts appear for this example because the return type of [insert] depends on the _value_ of the function's argument, a pattern that OCaml cannot handle. Since Coq's type system is much more expressive than OCaml's, such casts are unavoidable in general. Since the OCaml type-checker is no longer checking full safety of programs, we must rely on Coq's extractor to use casts only in provably safe ways. *)
adam@338 753 (* begin hide *)
adam@338 755 (* end hide *)
adamc@86 758 (** * A Certified Regular Expression Matcher *)
adamc@93 760 (** Another interesting example is regular expressions with dependent types that express which predicates over strings particular regexps implement. We can then assign a dependent type to a regular expression matching function, guaranteeing that it always decides the string property that we expect it to decide.
adam@425 762 Before defining the syntax of expressions, it is helpful to define an inductive type capturing the meaning of the Kleene star. That is, a string [s] matches regular expression [star e] if and only if [s] can be decomposed into a sequence of substrings that all match [e]. We use Coq's string support, which comes through a combination of the [String] library and some parsing notations built into Coq. Operators like [++] and functions like [length] that we know from lists are defined again for strings. Notation scopes help us control which versions we want to use in particular contexts.%\index{Vernacular commands!Open Scope}% *)
adamc@86 764 Require Import Ascii String.
adamc@91 768 Variable P : string -> Prop.
adamc@91 770 Inductive star : string -> Prop :=
adamc@91 771 | Empty : star ""
adamc@91 772 | Iter : forall s1 s2,
adamc@91 775 -> star (s1 ++ s2).
adam@480 778 (** Now we can make our first attempt at defining a [regexp] type that is indexed by predicates on strings, such that the index of a [regexp] tells us which language (string predicate) it recognizes. Here is a reasonable-looking definition that is restricted to constant characters and concatenation. We use the constructor [String], which is the analogue of list cons for the type [string], where [""] is like list nil.
adamc@93 780 Inductive regexp : (string -> Prop) -> Set :=
adamc@93 781 | Char : forall ch : ascii,
adamc@93 782 regexp (fun s => s = String ch "")
adamc@93 783 | Concat : forall (P1 P2 : string -> Prop) (r1 : regexp P1) (r2 : regexp P2),
adamc@93 784 regexp (fun s => exists s1, exists s2, s = s1 ++ s2 /\ P1 s1 /\ P2 s2).
adam@338 788 User error: Large non-propositional inductive types must be in Type
adam@454 791 What is a %\index{large inductive types}%large inductive type? In Coq, it is an inductive type that has a constructor that quantifies over some type of type [Type]. We have not worked with [Type] very much to this point. Every term of CIC has a type, including [Set] and [Prop], which are assigned type [Type]. The type [string -> Prop] from the failed definition also has type [Type].
adamc@93 793 It turns out that allowing large inductive types in [Set] leads to contradictions when combined with certain kinds of classical logic reasoning. Thus, by default, such types are ruled out. There is a simple fix for our [regexp] definition, which is to place our new type in [Type]. While fixing the problem, we also expand the list of constructors to cover the remaining regular expression operators. *)
adamc@89 795 Inductive regexp : (string -> Prop) -> Type :=
adamc@86 796 | Char : forall ch : ascii,
adamc@86 797 regexp (fun s => s = String ch "")
adamc@86 798 | Concat : forall P1 P2 (r1 : regexp P1) (r2 : regexp P2),
adamc@87 799 regexp (fun s => exists s1, exists s2, s = s1 ++ s2 /\ P1 s1 /\ P2 s2)
adamc@87 800 | Or : forall P1 P2 (r1 : regexp P1) (r2 : regexp P2),
adamc@91 801 regexp (fun s => P1 s \/ P2 s)
adamc@91 802 | Star : forall P (r : regexp P),
adam@425 805 (** Many theorems about strings are useful for implementing a certified regexp matcher, and few of them are in the [String] library. The book source includes statements, proofs, and hint commands for a handful of such omitted theorems. Since they are orthogonal to our use of dependent types, we hide them in the rendered versions of this book. *)
adamc@93 807 (* begin hide *)
adamc@86 810 Lemma length_emp : length "" <= 0.
adamc@86 814 Lemma append_emp : forall s, s = "" ++ s.
adamc@86 820 repeat match goal with
adamc@86 821 | [ |- context[match ?N with O => _ | S _ => _ end] ] => destruct N; crush
adamc@86 824 Lemma substring_le : forall s n m,
adamc@86 825 length (substring n m s) <= m.
adamc@86 829 Lemma substring_all : forall s,
adamc@86 830 substring 0 (length s) s = s.
adamc@86 834 Lemma substring_none : forall s n,
adamc@93 835 substring n 0 s = "".
adam@375 839 Hint Rewrite substring_all substring_none.
adamc@86 841 Lemma substring_split : forall s m,
adamc@86 842 substring 0 m s ++ substring m (length s - m) s = s.
adamc@86 846 Lemma length_app1 : forall s1 s2,
adamc@86 847 length s1 <= length (s1 ++ s2).
adamc@86 851 Hint Resolve length_emp append_emp substring_le substring_split length_app1.
adamc@86 853 Lemma substring_app_fst : forall s2 s1 n,
adamc@86 854 length s1 = n
adamc@86 855 -> substring 0 n (s1 ++ s2) = s1.
adamc@86 859 Lemma substring_app_snd : forall s2 s1 n,
adamc@86 860 length s1 = n
adamc@86 861 -> substring n (length (s1 ++ s2) - n) (s1 ++ s2) = s2.
adam@375 862 Hint Rewrite <- minus_n_O.
adam@375 867 Hint Rewrite substring_app_fst substring_app_snd using solve [trivial].
adamc@93 868 (* end hide *)
adamc@93 870 (** A few auxiliary functions help us in our final matcher definition. The function [split] will be used to implement the regexp concatenation case. *)
adamc@86 873 Variables P1 P2 : string -> Prop.
adamc@214 874 Variable P1_dec : forall s, {P1 s} + {~ P1 s}.
adamc@214 875 Variable P2_dec : forall s, {P2 s} + {~ P2 s}.
adamc@93 876 (** We require a choice of two arbitrary string predicates and functions for deciding them. *)
adamc@86 878 Variable s : string.
adamc@93 879 (** Our computation will take place relative to a single fixed string, so it is easiest to make it a [Variable], rather than an explicit argument to our functions. *)
adam@338 881 (** The function [split'] is the workhorse behind [split]. It searches through the possible ways of splitting [s] into two pieces, checking the two predicates against each such pair. The execution of [split'] progresses right-to-left, from splitting all of [s] into the first piece to splitting all of [s] into the second piece. It takes an extra argument, [n], which specifies how far along we are in this search process. *)
adam@297 883 Definition split' : forall n : nat, n <= length s
adamc@86 884 -> {exists s1, exists s2, length s1 <= n /\ s1 ++ s2 = s /\ P1 s1 /\ P2 s2}
adamc@214 885 + {forall s1 s2, length s1 <= n -> s1 ++ s2 = s -> ~ P1 s1 \/ ~ P2 s2}.
adamc@86 886 refine (fix F (n : nat) : n <= length s
adamc@86 887 -> {exists s1, exists s2, length s1 <= n /\ s1 ++ s2 = s /\ P1 s1 /\ P2 s2}
adamc@214 888 + {forall s1 s2, length s1 <= n -> s1 ++ s2 = s -> ~ P1 s1 \/ ~ P2 s2} :=
adamc@86 890 | O => fun _ => Reduce (P1_dec "" && P2_dec s)
adamc@93 891 | S n' => fun _ => (P1_dec (substring 0 (S n') s)
adamc@93 892 && P2_dec (substring (S n') (length s - S n') s))
adamc@86 893 || F n' _
adamc@86 894 end); clear F; crush; eauto 7;
adamc@86 896 | [ _ : length ?S <= 0 |- _ ] => destruct S
adam@338 897 | [ _ : length ?S' <= S ?N |- _ ] => destruct (eq_nat_dec (length S') (S N))
adam@338 901 (** There is one subtle point in the [split'] code that is worth mentioning. The main body of the function is a [match] on [n]. In the case where [n] is known to be [S n'], we write [S n'] in several places where we might be tempted to write [n]. However, without further work to craft proper [match] annotations, the type-checker does not use the equality between [n] and [S n']. Thus, it is common to see patterns repeated in [match] case bodies in dependently typed Coq code. We can at least use a [let] expression to avoid copying the pattern more than once, replacing the first case body with:
adamc@93 903 | S n' => fun _ => let n := S n' in
adamc@93 904 (P1_dec (substring 0 n s)
adamc@93 905 && P2_dec (substring n (length s - n) s))
adamc@93 906 || F n' _
adam@338 909 The [split] function itself is trivial to implement in terms of [split']. We just ask [split'] to begin its search with [n = length s]. *)
adamc@86 911 Definition split : {exists s1, exists s2, s = s1 ++ s2 /\ P1 s1 /\ P2 s2}
adamc@214 912 + {forall s1 s2, s = s1 ++ s2 -> ~ P1 s1 \/ ~ P2 s2}.
adamc@86 913 refine (Reduce (split' (n := length s) _)); crush; eauto.
adamc@86 917 Implicit Arguments split [P1 P2].
adamc@93 919 (* begin hide *)
adamc@91 920 Lemma app_empty_end : forall s, s ++ "" = s.
adamc@91 926 Lemma substring_self : forall s n,
adamc@91 928 -> substring n (length s - n) s = s.
adamc@91 932 Lemma substring_empty : forall s n m,
adamc@91 934 -> substring n m s = "".
adam@375 938 Hint Rewrite substring_self substring_empty using omega.
adamc@91 940 Lemma substring_split' : forall s n m,
adamc@91 941 substring n m s ++ substring (n + m) (length s - (n + m)) s
adamc@91 942 = substring n (length s - n) s.
adamc@91 948 Lemma substring_stack : forall s n2 m1 m2,
adamc@91 950 -> substring 0 m1 (substring n2 m2 s)
adamc@91 951 = substring n2 m1 s.
adamc@91 957 repeat match goal with
adamc@91 958 | [ |- context[match ?N with O => _ | S _ => _ end] ] => case_eq N; crush
adamc@91 961 Lemma substring_stack' : forall s n1 n2 m1 m2,
adamc@91 962 n1 + m1 <= m2
adamc@91 963 -> substring n1 m1 (substring n2 m2 s)
adamc@91 964 = substring (n1 + n2) m1 s.
adamc@91 967 | [ |- substring ?N1 _ _ = substring ?N2 _ _ ] =>
adamc@91 968 replace N1 with N2; crush
adamc@91 972 Lemma substring_suffix : forall s n,
adamc@91 973 n <= length s
adamc@91 974 -> length (substring n (length s - n) s) = length s - n.
adamc@91 978 Lemma substring_suffix_emp' : forall s n m,
adamc@91 979 substring n (S m) s = ""
adamc@91 980 -> n >= length s.
adamc@91 983 | [ |- ?N >= _ ] => destruct N; crush
adamc@91 986 [ |- S ?N >= S ?E ] => assert (N >= E); [ eauto | omega ]
adamc@91 990 Lemma substring_suffix_emp : forall s n m,
adamc@92 991 substring n m s = ""
adamc@92 992 -> m > 0
adamc@91 993 -> n >= length s.
adam@335 994 destruct m as [ | m]; [crush | intros; apply substring_suffix_emp' with m; assumption].
adamc@91 997 Hint Rewrite substring_stack substring_stack' substring_suffix
adamc@91 1000 Lemma minus_minus : forall n m1 m2,
adamc@91 1001 m1 + m2 <= n
adamc@91 1002 -> n - m1 - m2 = n - (m1 + m2).
adamc@91 1006 Lemma plus_n_Sm' : forall n m : nat, S (n + m) = m + S n.
adam@375 1010 Hint Rewrite minus_minus using omega.
adamc@93 1011 (* end hide *)
adamc@93 1013 (** One more helper function will come in handy: [dec_star], for implementing another linear search through ways of splitting a string, this time for implementing the Kleene star. *)
adamc@91 1016 Variable P : string -> Prop.
adamc@214 1017 Variable P_dec : forall s, {P s} + {~ P s}.
adam@338 1019 (** Some new lemmas and hints about the [star] type family are useful. We omit them here; they are included in the book source at this point. *)
adamc@93 1021 (* begin hide *)
adamc@91 1024 Lemma star_empty : forall s,
adamc@91 1025 length s = 0
adamc@91 1026 -> star P s.
adamc@91 1030 Lemma star_singleton : forall s, P s -> star P s.
adamc@91 1031 intros; rewrite <- (app_empty_end s); auto.
adamc@91 1034 Lemma star_app : forall s n m,
adamc@91 1035 P (substring n m s)
adamc@91 1036 -> star P (substring (n + m) (length s - (n + m)) s)
adamc@91 1037 -> star P (substring n (length s - n) s).
adamc@91 1040 | [ H : P (substring ?N ?M ?S) |- _ ] =>
adamc@91 1041 solve [ rewrite <- (substring_split S M); auto
adamc@91 1042 | rewrite <- (substring_split' S N M); auto ]
adamc@91 1046 Hint Resolve star_empty star_singleton star_app.
adamc@91 1048 Variable s : string.
adamc@91 1050 Lemma star_inv : forall s,
adamc@91 1052 -> s = ""
adamc@91 1053 \/ exists i, i < length s
adamc@91 1054 /\ P (substring 0 (S i) s)
adamc@91 1055 /\ star P (substring (S i) (length s - S i) s).
adamc@91 1056 Hint Extern 1 (exists i : nat, _) =>
adamc@91 1058 | [ H : P (String _ ?S) |- _ ] => exists (length S); crush
adamc@91 1063 | match goal with
adamc@91 1064 | [ _ : P ?S |- _ ] => destruct S; crush
adamc@91 1069 Lemma star_substring_inv : forall n,
adamc@91 1070 n <= length s
adamc@91 1071 -> star P (substring n (length s - n) s)
adamc@91 1072 -> substring n (length s - n) s = ""
adamc@91 1073 \/ exists l, l < length s - n
adamc@91 1074 /\ P (substring n (S l) s)
adamc@91 1075 /\ star P (substring (n + S l) (length s - (n + S l)) s).
adamc@91 1080 | [ H : star _ _ |- _ ] => generalize (star_inv H); do 3 crush; eauto
adamc@93 1083 (* end hide *)
adamc@93 1085 (** The function [dec_star''] implements a single iteration of the star. That is, it tries to find a string prefix matching [P], and it calls a parameter function on the remainder of the string. *)
adamc@91 1088 Variable n : nat.
adam@454 1089 (** Variable [n] is the length of the prefix of [s] that we have already processed. *)
adamc@91 1091 Variable P' : string -> Prop.
adamc@91 1092 Variable P'_dec : forall n' : nat, n' > n
adamc@91 1093 -> {P' (substring n' (length s - n') s)}
adamc@214 1094 + {~ P' (substring n' (length s - n') s)}.
adamc@93 1096 (** When we use [dec_star''], we will instantiate [P'_dec] with a function for continuing the search for more instances of [P] in [s]. *)
adamc@93 1098 (** Now we come to [dec_star''] itself. It takes as an input a natural [l] that records how much of the string has been searched so far, as we did for [split']. The return type expresses that [dec_star''] is looking for an index into [s] that splits [s] into a nonempty prefix and a suffix, such that the prefix satisfies [P] and the suffix satisfies [P']. *)
adam@297 1100 Definition dec_star'' : forall l : nat,
adam@297 1101 {exists l', S l' <= l
adamc@91 1102 /\ P (substring n (S l') s) /\ P' (substring (n + S l') (length s - (n + S l')) s)}
adamc@91 1103 + {forall l', S l' <= l
adamc@214 1104 -> ~ P (substring n (S l') s)
adamc@214 1105 \/ ~ P' (substring (n + S l') (length s - (n + S l')) s)}.
adamc@91 1106 refine (fix F (l : nat) : {exists l', S l' <= l
adam@480 1107 /\ P (substring n (S l') s) /\ P' (substring (n + S l') (length s - (n + S l')) s)}
adam@480 1108 + {forall l', S l' <= l
adam@480 1109 -> ~ P (substring n (S l') s)
adam@480 1110 \/ ~ P' (substring (n + S l') (length s - (n + S l')) s)} :=
adam@480 1112 | O => _
adam@480 1113 | S l' =>
adam@480 1114 (P_dec (substring n (S l') s) && P'_dec (n' := n + S l') _)
adam@480 1116 end); clear F; crush; eauto 7;
adam@480 1118 | [ H : ?X <= S ?Y |- _ ] => destruct (eq_nat_dec X (S Y)); crush
adamc@93 1123 (* begin hide *)
adamc@92 1124 Lemma star_length_contra : forall n,
adamc@92 1125 length s > n
adamc@92 1126 -> n >= length s
adamc@92 1131 Lemma star_length_flip : forall n n',
adamc@92 1132 length s - n <= S n'
adamc@92 1133 -> length s > n
adamc@92 1134 -> length s - n > 0.
adamc@92 1138 Hint Resolve star_length_contra star_length_flip substring_suffix_emp.
adamc@93 1139 (* end hide *)
adamc@93 1141 (** The work of [dec_star''] is nested inside another linear search by [dec_star'], which provides the final functionality we need, but for arbitrary suffixes of [s], rather than just for [s] overall. *)
adam@297 1143 Definition dec_star' : forall n n' : nat, length s - n' <= n
adamc@91 1144 -> {star P (substring n' (length s - n') s)}
adamc@214 1145 + {~ star P (substring n' (length s - n') s)}.
adamc@214 1146 refine (fix F (n n' : nat) : length s - n' <= n
adamc@91 1147 -> {star P (substring n' (length s - n') s)}
adamc@214 1148 + {~ star P (substring n' (length s - n') s)} :=
adamc@91 1150 | O => fun _ => Yes
adamc@91 1151 | S n'' => fun _ =>
adamc@91 1152 le_gt_dec (length s) n'
adam@338 1153 || dec_star'' (n := n') (star P)
adam@338 1154 (fun n0 _ => Reduce (F n'' n0 _)) (length s - n')
adamc@92 1155 end); clear F; crush; eauto;
adamc@92 1157 | [ H : star _ _ |- _ ] => apply star_substring_inv in H; crush; eauto
adamc@92 1160 | [ H1 : _ < _ - _, H2 : forall l' : nat, _ <= _ - _ -> _ |- _ ] =>
adamc@92 1161 generalize (H2 _ (lt_le_S _ _ H1)); tauto
adam@380 1165 (** Finally, we have [dec_star], defined by straightforward reduction from [dec_star']. *)
adamc@214 1167 Definition dec_star : {star P s} + {~ star P s}.
adam@380 1168 refine (Reduce (dec_star' (n := length s) 0 _)); crush.
adamc@93 1172 (* begin hide *)
adamc@86 1173 Lemma app_cong : forall x1 y1 x2 y2,
adamc@86 1175 -> y1 = y2
adamc@86 1176 -> x1 ++ y1 = x2 ++ y2.
adamc@93 1181 (* end hide *)
adamc@93 1183 (** With these helper functions completed, the implementation of our [matches] function is refreshingly straightforward. We only need one small piece of specific tactic work beyond what [crush] does for us. *)
adam@297 1185 Definition matches : forall P (r : regexp P) s, {P s} + {~ P s}.
adamc@214 1186 refine (fix F P (r : regexp P) s : {P s} + {~ P s} :=
adamc@86 1188 | Char ch => string_dec s (String ch "")
adamc@86 1189 | Concat _ _ r1 r2 => Reduce (split (F _ r1) (F _ r2) s)
adamc@87 1190 | Or _ _ r1 r2 => F _ r1 s || F _ r2 s
adamc@91 1191 | Star _ r => dec_star _ _ _
adam@426 1194 | [ H : _ |- _ ] => generalize (H _ _ (eq_refl _))
adam@283 1198 (** It is interesting to pause briefly to consider alternate implementations of [matches]. Dependent types give us much latitude in how specific correctness properties may be encoded with types. For instance, we could have made [regexp] a non-indexed inductive type, along the lines of what is possible in traditional ML and Haskell. We could then have implemented a recursive function to map [regexp]s to their intended meanings, much as we have done with types and programs in other examples. That style is compatible with the [refine]-based approach that we have used here, and it might be an interesting exercise to redo the code from this subsection in that alternate style or some further encoding of the reader's choice. The main advantage of indexed inductive types is that they generally lead to the smallest amount of code. *)
adamc@93 1200 (* begin hide *)
adamc@86 1201 Example hi := Concat (Char "h"%char) (Char "i"%char).
adam@380 1202 Eval hnf in matches hi "hi".
adam@380 1203 Eval hnf in matches hi "bye".
adamc@87 1205 Example a_b := Or (Char "a"%char) (Char "b"%char).
adam@380 1206 Eval hnf in matches a_b "".
adam@380 1207 Eval hnf in matches a_b "a".
adam@380 1208 Eval hnf in matches a_b "aa".
adam@380 1209 Eval hnf in matches a_b "b".
adam@283 1210 (* end hide *)