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author Adam Chlipala <adam@chlipala.net>
date Wed, 29 Aug 2012 18:26:26 -0400
parents 49bd155dfc52
children 1fd4109f7b31
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adam@371 1 (* Copyright (c) 2008-2012, Adam Chlipala
adamc@83 2 *
adamc@83 3 * This work is licensed under a
adamc@83 4 * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
adamc@83 5 * Unported License.
adamc@83 6 * The license text is available at:
adamc@83 7 * http://creativecommons.org/licenses/by-nc-nd/3.0/
adamc@83 8 *)
adamc@83 9
adamc@83 10 (* begin hide *)
adamc@85 11 Require Import Arith Bool List.
adamc@83 12
adam@314 13 Require Import CpdtTactics MoreSpecif.
adamc@83 14
adamc@83 15 Set Implicit Arguments.
adamc@83 16 (* end hide *)
adamc@83 17
adamc@83 18
adamc@83 19 (** %\chapter{More Dependent Types}% *)
adamc@83 20
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.
adamc@83 22
adam@338 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 (that is, dependent inductive types outside [Prop]), a possibility that sets Coq apart from all of the competition not based on type theory. *)
adamc@83 24
adamc@84 25
adamc@84 26 (** * Length-Indexed Lists *)
adamc@84 27
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 29
adamc@84 30 Section ilist.
adamc@84 31 Variable A : Set.
adamc@84 32
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 36
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.
adamc@84 38
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@84 40
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@84 42
adamc@213 43 Fixpoint app n1 (ls1 : ilist n1) n2 (ls2 : ilist n2) : ilist (n1 + n2) :=
adamc@213 44 match ls1 with
adamc@213 45 | Nil => ls2
adamc@213 46 | Cons _ x ls1' => Cons x (app ls1' ls2)
adamc@213 47 end.
adamc@84 48
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 50
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)
adam@338 56 end.
adamc@100 57 (* end thide *)
adamc@84 58
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.
adamc@84 60
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.
adamc@84 62
adam@425 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. This stratification between a compile-time universe and a run-time universe, with no references to the latter in the former, gives rise to the terminology "stratified." Our next example would be harder to implement in a stratified system. We write an injection function from regular lists to length-indexed lists. A stratified implementation would need to duplicate the definition of lists across compile-time and run-time versions, and the run-time versions would need to be indexed by the compile-time versions. *)
adamc@84 64
adamc@100 65 (* EX: Implement injection from normal lists *)
adamc@100 66
adamc@100 67 (* begin thide *)
adam@454 68 Fixpoint inject (ls : list A) : ilist (length ls) :=
adam@454 69 match ls with
adam@454 70 | nil => Nil
adam@454 71 | h :: t => Cons h (inject t)
adam@454 72 end.
adamc@84 73
adamc@84 74 (** We can define an inverse conversion and prove that it really is an inverse. *)
adamc@84 75
adam@454 76 Fixpoint unject n (ls : ilist n) : list A :=
adam@454 77 match ls with
adam@454 78 | Nil => nil
adam@454 79 | Cons _ h t => h :: unject t
adam@454 80 end.
adamc@84 81
adam@454 82 Theorem inject_inverse : forall ls, unject (inject ls) = ls.
adam@454 83 induction ls; crush.
adam@454 84 Qed.
adamc@100 85 (* end thide *)
adamc@100 86
adam@338 87 (* EX: Implement statically checked "car"/"hd" *)
adamc@84 88
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.
adamc@84 90 [[
adam@454 91 Definition hd n (ls : ilist (S n)) : A :=
adam@454 92 match ls with
adam@454 93 | Nil => ???
adam@454 94 | Cons _ h _ => h
adam@454 95 end.
adamc@213 96 ]]
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:
adamc@84 98 [[
adam@454 99 Definition hd n (ls : ilist (S n)) : A :=
adam@454 100 match ls with
adam@454 101 | Cons _ h _ => h
adam@454 102 end.
adam@338 103 ]]
adamc@84 104
adam@338 105 <<
adamc@84 106 Error: Non exhaustive pattern-matching: no clause found for pattern Nil
adam@338 107 >>
adamc@84 108
adam@425 109 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. It is educational to discover that encoding ourselves directly. We might try using an [in] clause somehow.
adamc@84 110
adamc@84 111 [[
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@454 115 end.
adamc@84 116 ]]
adamc@84 117
adam@338 118 <<
adam@338 119 Error: The reference n was not found in the current environment
adam@338 120 >>
adam@338 121
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@84 123
adamc@84 124 Our final, working attempt at [hd] uses an auxiliary function and a surprising [return] annotation. *)
adamc@84 125
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@454 131 end.
adamc@84 132
adam@454 133 Check hd'.
adam@283 134 (** %\vspace{-.15in}% [[
adam@283 135 hd'
adam@283 136 : forall n : nat, ilist n -> match n with
adam@283 137 | 0 => unit
adam@283 138 | S _ => A
adam@283 139 end
adam@302 140 ]]
adam@302 141 *)
adam@283 142
adam@454 143 Definition hd n (ls : ilist (S n)) : A := hd' ls.
adamc@100 144 (* end thide *)
adamc@84 145
adam@338 146 End ilist.
adam@338 147
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]. *)
adamc@84 149
adamc@85 150
adam@371 151 (** * The One Rule of Dependent Pattern Matching in Coq *)
adam@371 152
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@371 154
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@371 156
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@371 158
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@371 160 [[
adam@371 161 match E in (T x1 ... xn) as y return U with
adam@371 162 | C z1 ... zm => B
adam@371 163 | ...
adam@371 164 end
adam@371 165 ]]
adam@371 166
adam@371 167 The discriminee is a term [E], a value in some inductive type family [T], which takes [n] arguments. An %\index{in clause}%[in] clause binds an explicit name [xi] for the [i]th argument passed to [T] in the type of [E]. An %\index{as clause}%[as] clause binds the name [y] to refer to the discriminee [E].
adam@371 168
adam@371 169 We bind these new variables [xi] and [y] 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@371 170
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@371 172
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@371 174
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@371 176
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.) *)
adam@371 178
adam@371 179
adamc@85 180 (** * A Tagless Interpreter *)
adamc@85 181
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@85 183
adamc@85 184 Inductive type : Set :=
adamc@85 185 | Nat : type
adamc@85 186 | Bool : type
adamc@85 187 | Prod : type -> type -> type.
adamc@85 188
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@85 193
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@85 197
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.
adamc@85 201
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.
adamc@85 203
adam@398 204 We can give types and expressions semantics in a new style, based critically on the chance for _type-level computation_. *)
adamc@85 205
adamc@85 206 Fixpoint typeDenote (t : type) : Set :=
adamc@85 207 match t with
adamc@85 208 | Nat => nat
adamc@85 209 | Bool => bool
adamc@85 210 | Prod t1 t2 => typeDenote t1 * typeDenote t2
adamc@85 211 end%type.
adamc@85 212
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@85 214
adamc@85 215 We can define a function [expDenote] that is typed in terms of [typeDenote]. *)
adamc@85 216
adamc@213 217 Fixpoint expDenote t (e : exp t) : typeDenote t :=
adamc@213 218 match e with
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@85 222
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@85 226
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')
adamc@85 230 end.
adamc@85 231
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@437 237
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@85 239
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@85 241 [[
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@85 246 end.
adam@338 247 ]]
adamc@85 248
adam@338 249 <<
adamc@85 250 Error: The reference t2 was not found in the current environment
adam@338 251 >>
adamc@85 252
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@85 254
adamc@100 255 (* EX: Define a function [pairOut : forall t1 t2, exp (Prod t1 t2) -> option (exp t1 * exp t2)] *)
adamc@100 256
adamc@100 257 (* begin thide *)
adamc@85 258 Definition pairOutType (t : type) :=
adamc@85 259 match t with
adamc@85 260 | Prod t1 t2 => option (exp t1 * exp t2)
adamc@85 261 | _ => unit
adamc@85 262 end.
adamc@85 263
adamc@85 264 (** When passed a type that is a product, [pairOutType] returns our final desired type. On any other input type, [pairOutType] returns [unit], since we do not care about extracting components of non-pairs. Now we can write another helper function to provide the default behavior of [pairOut], which we will apply for inputs that are not literal pairs. *)
adamc@85 265
adamc@85 266 Definition pairOutDefault (t : type) :=
adamc@85 267 match t return (pairOutType t) with
adamc@85 268 | Prod _ _ => None
adamc@85 269 | _ => tt
adamc@85 270 end.
adamc@85 271
adamc@85 272 (** Now [pairOut] is deceptively easy to write. *)
adamc@85 273
adamc@85 274 Definition pairOut t (e : exp t) :=
adamc@85 275 match e in (exp t) return (pairOutType t) with
adamc@85 276 | Pair _ _ e1 e2 => Some (e1, e2)
adamc@85 277 | _ => pairOutDefault _
adamc@85 278 end.
adamc@100 279 (* end thide *)
adamc@85 280
adam@338 281 (** There is one important subtlety in this definition. Coq allows us to use convenient ML-style pattern matching notation, but, internally and in proofs, we see that patterns are expanded out completely, matching one level of inductive structure at a time. Thus, the default case in the [match] above expands out to one case for each constructor of [exp] besides [Pair], and the underscore in [pairOutDefault _] is resolved differently in each case. From an ML or Haskell programmer's perspective, what we have here is type inference determining which code is run (returning either [None] or [tt]), which goes beyond what is possible with type inference guiding parametric polymorphism in Hindley-Milner languages%\index{Hindley-Milner}%, but is similar to what goes on with Haskell type classes%\index{type classes}%.
adamc@85 282
adam@454 283 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@85 284
adamc@204 285 Fixpoint cfold t (e : exp t) : exp t :=
adamc@204 286 match e with
adamc@85 287 | NConst n => NConst n
adamc@85 288 | Plus e1 e2 =>
adamc@85 289 let e1' := cfold e1 in
adamc@85 290 let e2' := cfold e2 in
adam@417 291 match e1', e2' return exp Nat with
adamc@85 292 | NConst n1, NConst n2 => NConst (n1 + n2)
adamc@85 293 | _, _ => Plus e1' e2'
adamc@85 294 end
adamc@85 295 | Eq e1 e2 =>
adamc@85 296 let e1' := cfold e1 in
adamc@85 297 let e2' := cfold e2 in
adam@417 298 match e1', e2' return exp Bool with
adamc@85 299 | NConst n1, NConst n2 => BConst (if eq_nat_dec n1 n2 then true else false)
adamc@85 300 | _, _ => Eq e1' e2'
adamc@85 301 end
adamc@85 302
adamc@85 303 | BConst b => BConst b
adamc@85 304 | And e1 e2 =>
adamc@85 305 let e1' := cfold e1 in
adamc@85 306 let e2' := cfold e2 in
adam@417 307 match e1', e2' return exp Bool with
adamc@85 308 | BConst b1, BConst b2 => BConst (b1 && b2)
adamc@85 309 | _, _ => And e1' e2'
adamc@85 310 end
adamc@85 311 | If _ e e1 e2 =>
adamc@85 312 let e' := cfold e in
adamc@85 313 match e' with
adamc@85 314 | BConst true => cfold e1
adamc@85 315 | BConst false => cfold e2
adamc@85 316 | _ => If e' (cfold e1) (cfold e2)
adamc@85 317 end
adamc@85 318
adamc@85 319 | Pair _ _ e1 e2 => Pair (cfold e1) (cfold e2)
adamc@85 320 | Fst _ _ e =>
adamc@85 321 let e' := cfold e in
adamc@85 322 match pairOut e' with
adamc@85 323 | Some p => fst p
adamc@85 324 | None => Fst e'
adamc@85 325 end
adamc@85 326 | Snd _ _ e =>
adamc@85 327 let e' := cfold e in
adamc@85 328 match pairOut e' with
adamc@85 329 | Some p => snd p
adamc@85 330 | None => Snd e'
adamc@85 331 end
adamc@85 332 end.
adamc@85 333
adamc@85 334 (** The correctness theorem for [cfold] turns out to be easy to prove, once we get over one serious hurdle. *)
adamc@85 335
adamc@85 336 Theorem cfold_correct : forall t (e : exp t), expDenote e = expDenote (cfold e).
adamc@100 337 (* begin thide *)
adamc@85 338 induction e; crush.
adamc@85 339
adamc@85 340 (** The first remaining subgoal is:
adamc@85 341
adamc@85 342 [[
adamc@85 343 expDenote (cfold e1) + expDenote (cfold e2) =
adamc@85 344 expDenote
adamc@85 345 match cfold e1 with
adamc@85 346 | NConst n1 =>
adamc@85 347 match cfold e2 with
adamc@85 348 | NConst n2 => NConst (n1 + n2)
adamc@85 349 | Plus _ _ => Plus (cfold e1) (cfold e2)
adamc@85 350 | Eq _ _ => Plus (cfold e1) (cfold e2)
adamc@85 351 | BConst _ => Plus (cfold e1) (cfold e2)
adamc@85 352 | And _ _ => Plus (cfold e1) (cfold e2)
adamc@85 353 | If _ _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 354 | Pair _ _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 355 | Fst _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 356 | Snd _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 357 end
adamc@85 358 | Plus _ _ => Plus (cfold e1) (cfold e2)
adamc@85 359 | Eq _ _ => Plus (cfold e1) (cfold e2)
adamc@85 360 | BConst _ => Plus (cfold e1) (cfold e2)
adamc@85 361 | And _ _ => Plus (cfold e1) (cfold e2)
adamc@85 362 | If _ _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 363 | Pair _ _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 364 | Fst _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 365 | Snd _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 366 end
adamc@213 367
adamc@85 368 ]]
adamc@85 369
adam@454 370 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@85 371 [[
adamc@85 372 destruct (cfold e1).
adam@338 373 ]]
adamc@85 374
adam@338 375 <<
adamc@85 376 User error: e1 is used in hypothesis e
adam@338 377 >>
adamc@85 378
adamc@85 379 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.
adamc@85 380
adam@350 381 For our current proof, we can use a tactic [dep_destruct]%\index{tactics!dep\_destruct}% defined in the book [CpdtTactics] module. General elimination/inversion of dependently typed hypotheses is undecidable, since it must be implemented with [match] expressions that have the restriction on [in] clauses that we have already discussed. 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 [dep_destruct]'s implementation in Ltac, 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@85 382
adamc@85 383 dep_destruct (cfold e1).
adamc@85 384
adamc@85 385 (** 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].
adamc@85 386
adam@405 387 This is the only new trick we need to learn to complete the proof. We can back up and give a short, automated proof. *)
adamc@85 388
adamc@85 389 Restart.
adamc@85 390
adamc@85 391 induction e; crush;
adamc@85 392 repeat (match goal with
adam@405 393 | [ |- context[match cfold ?E with NConst _ => _ | _ => _ end] ] =>
adamc@213 394 dep_destruct (cfold E)
adamc@213 395 | [ |- context[match pairOut (cfold ?E) with Some _ => _
adamc@213 396 | None => _ end] ] =>
adamc@213 397 dep_destruct (cfold E)
adamc@85 398 | [ |- (if ?E then _ else _) = _ ] => destruct E
adamc@85 399 end; crush).
adamc@85 400 Qed.
adamc@100 401 (* end thide *)
adamc@86 402
adam@405 403 (** With this example, we get a first taste of how to build automated proofs that adapt automatically to changes in function definitions. *)
adam@405 404
adamc@86 405
adam@338 406 (** * Dependently Typed Red-Black Trees *)
adamc@94 407
adam@338 408 (** Red-black trees are a favorite purely functional data structure with an interesting invariant. We can use dependent types to enforce that operations on red-black trees preserve the invariant. For simplicity, we specialize our red-black trees to represent sets of [nat]s. *)
adamc@100 409
adamc@94 410 Inductive color : Set := Red | Black.
adamc@94 411
adamc@94 412 Inductive rbtree : color -> nat -> Set :=
adamc@94 413 | Leaf : rbtree Black 0
adamc@214 414 | RedNode : forall n, rbtree Black n -> nat -> rbtree Black n -> rbtree Red n
adamc@94 415 | BlackNode : forall c1 c2 n, rbtree c1 n -> nat -> rbtree c2 n -> rbtree Black (S n).
adamc@94 416
adamc@214 417 (** A value of type [rbtree c d] is a red-black tree node whose root has color [c] and that has black depth [d]. The latter property means that there are no more than [d] black-colored nodes on any path from the root to a leaf. *)
adamc@214 418
adamc@214 419 (** 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@214 420
adamc@100 421 (* EX: Prove that every [rbtree] is balanced. *)
adamc@100 422
adamc@100 423 (* begin thide *)
adamc@95 424 Require Import Max Min.
adamc@95 425
adamc@95 426 Section depth.
adamc@95 427 Variable f : nat -> nat -> nat.
adamc@95 428
adamc@214 429 Fixpoint depth c n (t : rbtree c n) : nat :=
adamc@95 430 match t with
adamc@95 431 | Leaf => 0
adamc@95 432 | RedNode _ t1 _ t2 => S (f (depth t1) (depth t2))
adamc@95 433 | BlackNode _ _ _ t1 _ t2 => S (f (depth t1) (depth t2))
adamc@95 434 end.
adamc@95 435 End depth.
adamc@95 436
adam@338 437 (** 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]. *)
adamc@214 438
adam@283 439 Check min_dec.
adam@283 440 (** %\vspace{-.15in}% [[
adam@283 441 min_dec
adam@283 442 : forall n m : nat, {min n m = n} + {min n m = m}
adam@302 443 ]]
adam@302 444 *)
adam@283 445
adamc@95 446 Theorem depth_min : forall c n (t : rbtree c n), depth min t >= n.
adamc@95 447 induction t; crush;
adamc@95 448 match goal with
adamc@95 449 | [ |- context[min ?X ?Y] ] => destruct (min_dec X Y)
adamc@95 450 end; crush.
adamc@95 451 Qed.
adamc@95 452
adamc@214 453 (** There is an analogous upper-bound theorem based on black depth. Unfortunately, a symmetric proof script does not suffice to establish it. *)
adamc@214 454
adamc@214 455 Theorem depth_max : forall c n (t : rbtree c n), depth max t <= 2 * n + 1.
adamc@214 456 induction t; crush;
adamc@214 457 match goal with
adamc@214 458 | [ |- context[max ?X ?Y] ] => destruct (max_dec X Y)
adamc@214 459 end; crush.
adamc@214 460
adamc@214 461 (** Two subgoals remain. One of them is: [[
adamc@214 462 n : nat
adamc@214 463 t1 : rbtree Black n
adamc@214 464 n0 : nat
adamc@214 465 t2 : rbtree Black n
adamc@214 466 IHt1 : depth max t1 <= n + (n + 0) + 1
adamc@214 467 IHt2 : depth max t2 <= n + (n + 0) + 1
adamc@214 468 e : max (depth max t1) (depth max t2) = depth max t1
adamc@214 469 ============================
adamc@214 470 S (depth max t1) <= n + (n + 0) + 1
adamc@214 471
adamc@214 472 ]]
adamc@214 473
adam@398 474 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@214 475
adamc@214 476 Abort.
adamc@214 477
adamc@214 478 (** 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@214 479
adamc@95 480 Lemma depth_max' : forall c n (t : rbtree c n), match c with
adamc@95 481 | Red => depth max t <= 2 * n + 1
adamc@95 482 | Black => depth max t <= 2 * n
adamc@95 483 end.
adamc@95 484 induction t; crush;
adamc@95 485 match goal with
adamc@95 486 | [ |- context[max ?X ?Y] ] => destruct (max_dec X Y)
adamc@100 487 end; crush;
adamc@100 488 repeat (match goal with
adamc@214 489 | [ H : context[match ?C with Red => _ | Black => _ end] |- _ ] =>
adamc@214 490 destruct C
adamc@100 491 end; crush).
adamc@95 492 Qed.
adamc@95 493
adam@338 494 (** 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@214 495
adamc@95 496 Theorem depth_max : forall c n (t : rbtree c n), depth max t <= 2 * n + 1.
adamc@95 497 intros; generalize (depth_max' t); destruct c; crush.
adamc@95 498 Qed.
adamc@95 499
adamc@214 500 (** 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@214 501
adamc@95 502 Theorem balanced : forall c n (t : rbtree c n), 2 * depth min t + 1 >= depth max t.
adamc@95 503 intros; generalize (depth_min t); generalize (depth_max t); crush.
adamc@95 504 Qed.
adamc@100 505 (* end thide *)
adamc@95 506
adamc@214 507 (** 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@95 508
adamc@94 509 Inductive rtree : nat -> Set :=
adamc@94 510 | RedNode' : forall c1 c2 n, rbtree c1 n -> nat -> rbtree c2 n -> rtree n.
adamc@94 511
adam@338 512 (** 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@214 513
adamc@96 514 Section present.
adamc@96 515 Variable x : nat.
adamc@96 516
adamc@214 517 Fixpoint present c n (t : rbtree c n) : Prop :=
adamc@96 518 match t with
adamc@96 519 | Leaf => False
adamc@96 520 | RedNode _ a y b => present a \/ x = y \/ present b
adamc@96 521 | BlackNode _ _ _ a y b => present a \/ x = y \/ present b
adamc@96 522 end.
adamc@96 523
adamc@96 524 Definition rpresent n (t : rtree n) : Prop :=
adamc@96 525 match t with
adamc@96 526 | RedNode' _ _ _ a y b => present a \/ x = y \/ present b
adamc@96 527 end.
adamc@96 528 End present.
adamc@96 529
adam@338 530 (** 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@214 531
adamc@100 532 Locate "{ _ : _ & _ }".
adam@443 533 (** %\vspace{-.15in}%[[
adamc@214 534 Notation Scope
adamc@214 535 "{ x : A & P }" := sigT (fun x : A => P)
adam@302 536 ]]
adam@302 537 *)
adamc@214 538
adamc@100 539 Print sigT.
adam@443 540 (** %\vspace{-.15in}%[[
adamc@214 541 Inductive sigT (A : Type) (P : A -> Type) : Type :=
adamc@214 542 existT : forall x : A, P x -> sigT P
adam@302 543 ]]
adam@302 544 *)
adamc@214 545
adamc@214 546 (** It will be helpful to define a concise notation for the constructor of [sigT]. *)
adamc@100 547
adamc@94 548 Notation "{< x >}" := (existT _ _ x).
adamc@94 549
adamc@214 550 (** 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.
adamc@214 551
adam@338 552 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@338 553
adam@425 554 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@214 555
adamc@94 556 Definition balance1 n (a : rtree n) (data : nat) c2 :=
adamc@214 557 match a in rtree n return rbtree c2 n
adamc@214 558 -> { c : color & rbtree c (S n) } with
adam@380 559 | RedNode' _ c0 _ t1 y t2 =>
adam@380 560 match t1 in rbtree c n return rbtree c0 n -> rbtree c2 n
adamc@214 561 -> { c : color & rbtree c (S n) } with
adamc@214 562 | RedNode _ a x b => fun c d =>
adamc@214 563 {<RedNode (BlackNode a x b) y (BlackNode c data d)>}
adamc@94 564 | t1' => fun t2 =>
adam@380 565 match t2 in rbtree c n return rbtree Black n -> rbtree c2 n
adamc@214 566 -> { c : color & rbtree c (S n) } with
adamc@214 567 | RedNode _ b x c => fun a d =>
adamc@214 568 {<RedNode (BlackNode a y b) x (BlackNode c data d)>}
adamc@95 569 | b => fun a t => {<BlackNode (RedNode a y b) data t>}
adamc@94 570 end t1'
adamc@94 571 end t2
adamc@94 572 end.
adamc@94 573
adam@405 574 (** 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.
adamc@214 575
adam@425 576 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.
adamc@214 577
adam@338 578 Here is the symmetric function [balance2], for cases where the possibly invalid tree appears on the right rather than on the left. *)
adamc@214 579
adamc@94 580 Definition balance2 n (a : rtree n) (data : nat) c2 :=
adamc@94 581 match a in rtree n return rbtree c2 n -> { c : color & rbtree c (S n) } with
adam@380 582 | RedNode' _ c0 _ t1 z t2 =>
adam@380 583 match t1 in rbtree c n return rbtree c0 n -> rbtree c2 n
adamc@214 584 -> { c : color & rbtree c (S n) } with
adamc@214 585 | RedNode _ b y c => fun d a =>
adamc@214 586 {<RedNode (BlackNode a data b) y (BlackNode c z d)>}
adamc@94 587 | t1' => fun t2 =>
adam@380 588 match t2 in rbtree c n return rbtree Black n -> rbtree c2 n
adamc@214 589 -> { c : color & rbtree c (S n) } with
adamc@214 590 | RedNode _ c z' d => fun b a =>
adamc@214 591 {<RedNode (BlackNode a data b) z (BlackNode c z' d)>}
adamc@95 592 | b => fun a t => {<BlackNode t data (RedNode a z b)>}
adamc@94 593 end t1'
adamc@94 594 end t2
adamc@94 595 end.
adamc@94 596
adamc@214 597 (** 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@214 598
adamc@94 599 Section insert.
adamc@94 600 Variable x : nat.
adamc@94 601
adamc@214 602 (** 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@214 603
adamc@94 604 Definition insResult c n :=
adamc@94 605 match c with
adamc@94 606 | Red => rtree n
adamc@94 607 | Black => { c' : color & rbtree c' n }
adamc@94 608 end.
adamc@94 609
adam@338 610 (** 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@214 611
adamc@214 612 Here is the definition of [ins]. Again, we do not want to dwell on the functional details. *)
adamc@214 613
adamc@214 614 Fixpoint ins c n (t : rbtree c n) : insResult c n :=
adamc@214 615 match t with
adamc@94 616 | Leaf => {< RedNode Leaf x Leaf >}
adamc@94 617 | RedNode _ a y b =>
adamc@94 618 if le_lt_dec x y
adamc@94 619 then RedNode' (projT2 (ins a)) y b
adamc@94 620 else RedNode' a y (projT2 (ins b))
adamc@94 621 | BlackNode c1 c2 _ a y b =>
adamc@94 622 if le_lt_dec x y
adamc@94 623 then
adamc@94 624 match c1 return insResult c1 _ -> _ with
adamc@94 625 | Red => fun ins_a => balance1 ins_a y b
adamc@94 626 | _ => fun ins_a => {< BlackNode (projT2 ins_a) y b >}
adamc@94 627 end (ins a)
adamc@94 628 else
adamc@94 629 match c2 return insResult c2 _ -> _ with
adamc@94 630 | Red => fun ins_b => balance2 ins_b y a
adamc@94 631 | _ => fun ins_b => {< BlackNode a y (projT2 ins_b) >}
adamc@94 632 end (ins b)
adamc@94 633 end.
adamc@94 634
adam@398 635 (** 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 naively 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@214 636
adamc@214 637 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@214 638
adamc@94 639 Definition insertResult c n :=
adamc@94 640 match c with
adamc@94 641 | Red => rbtree Black (S n)
adamc@94 642 | Black => { c' : color & rbtree c' n }
adamc@94 643 end.
adamc@94 644
adamc@214 645 (** A simple clean-up procedure translates [insResult]s into [insertResult]s. *)
adamc@214 646
adamc@97 647 Definition makeRbtree c n : insResult c n -> insertResult c n :=
adamc@214 648 match c with
adamc@94 649 | Red => fun r =>
adamc@214 650 match r with
adamc@94 651 | RedNode' _ _ _ a x b => BlackNode a x b
adamc@94 652 end
adamc@94 653 | Black => fun r => r
adamc@94 654 end.
adamc@94 655
adamc@214 656 (** 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@214 657
adamc@97 658 Implicit Arguments makeRbtree [c n].
adamc@94 659
adamc@214 660 (** Finally, we define [insert] as a simple composition of [ins] and [makeRbtree]. *)
adamc@214 661
adamc@94 662 Definition insert c n (t : rbtree c n) : insertResult c n :=
adamc@97 663 makeRbtree (ins t).
adamc@94 664
adamc@214 665 (** 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@214 666
adamc@95 667 Section present.
adamc@95 668 Variable z : nat.
adamc@95 669
adamc@214 670 (** 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.
adamc@214 671
adam@367 672 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@214 673
adamc@98 674 Ltac present_balance :=
adamc@98 675 crush;
adamc@98 676 repeat (match goal with
adam@425 677 | [ _ : context[match ?T with Leaf => _ | _ => _ end] |- _ ] =>
adam@425 678 dep_destruct T
adam@405 679 | [ |- context[match ?T with Leaf => _ | _ => _ end] ] => dep_destruct T
adamc@98 680 end; crush).
adamc@98 681
adamc@214 682 (** The balance correctness theorems are simple first-order logic equivalences, where we use the function [projT2] to project the payload of a [sigT] value. *)
adamc@214 683
adam@294 684 Lemma present_balance1 : forall n (a : rtree n) (y : nat) c2 (b : rbtree c2 n),
adamc@95 685 present z (projT2 (balance1 a y b))
adamc@95 686 <-> rpresent z a \/ z = y \/ present z b.
adamc@98 687 destruct a; present_balance.
adamc@95 688 Qed.
adamc@95 689
adamc@213 690 Lemma present_balance2 : forall n (a : rtree n) (y : nat) c2 (b : rbtree c2 n),
adamc@95 691 present z (projT2 (balance2 a y b))
adamc@95 692 <-> rpresent z a \/ z = y \/ present z b.
adamc@98 693 destruct a; present_balance.
adamc@95 694 Qed.
adamc@95 695
adamc@214 696 (** 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@214 697
adamc@95 698 Definition present_insResult c n :=
adamc@95 699 match c return (rbtree c n -> insResult c n -> Prop) with
adamc@95 700 | Red => fun t r => rpresent z r <-> z = x \/ present z t
adamc@95 701 | Black => fun t r => present z (projT2 r) <-> z = x \/ present z t
adamc@95 702 end.
adamc@95 703
adamc@214 704 (** 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@214 705
adamc@95 706 Theorem present_ins : forall c n (t : rbtree c n),
adamc@95 707 present_insResult t (ins t).
adamc@95 708 induction t; crush;
adamc@95 709 repeat (match goal with
adam@338 710 | [ _ : context[if ?E then _ else _] |- _ ] => destruct E
adamc@95 711 | [ |- context[if ?E then _ else _] ] => destruct E
adam@338 712 | [ _ : context[match ?C with Red => _ | Black => _ end]
adamc@214 713 |- _ ] => destruct C
adamc@95 714 end; crush);
adamc@95 715 try match goal with
adam@338 716 | [ _ : context[balance1 ?A ?B ?C] |- _ ] =>
adamc@95 717 generalize (present_balance1 A B C)
adamc@95 718 end;
adamc@95 719 try match goal with
adam@338 720 | [ _ : context[balance2 ?A ?B ?C] |- _ ] =>
adamc@95 721 generalize (present_balance2 A B C)
adamc@95 722 end;
adamc@95 723 try match goal with
adamc@95 724 | [ |- context[balance1 ?A ?B ?C] ] =>
adamc@95 725 generalize (present_balance1 A B C)
adamc@95 726 end;
adamc@95 727 try match goal with
adamc@95 728 | [ |- context[balance2 ?A ?B ?C] ] =>
adamc@95 729 generalize (present_balance2 A B C)
adamc@95 730 end;
adamc@214 731 crush;
adamc@95 732 match goal with
adamc@95 733 | [ z : nat, x : nat |- _ ] =>
adamc@95 734 match goal with
adamc@95 735 | [ H : z = x |- _ ] => rewrite H in *; clear H
adamc@95 736 end
adamc@95 737 end;
adamc@95 738 tauto.
adamc@95 739 Qed.
adamc@95 740
adamc@214 741 (** 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@214 742
adamc@213 743 Ltac present_insert :=
adamc@213 744 unfold insert; intros n t; inversion t;
adamc@97 745 generalize (present_ins t); simpl;
adamc@97 746 dep_destruct (ins t); tauto.
adamc@97 747
adamc@95 748 Theorem present_insert_Red : forall n (t : rbtree Red n),
adamc@95 749 present z (insert t)
adamc@95 750 <-> (z = x \/ present z t).
adamc@213 751 present_insert.
adamc@95 752 Qed.
adamc@95 753
adamc@95 754 Theorem present_insert_Black : forall n (t : rbtree Black n),
adamc@95 755 present z (projT2 (insert t))
adamc@95 756 <-> (z = x \/ present z t).
adamc@213 757 present_insert.
adamc@95 758 Qed.
adamc@95 759 End present.
adamc@94 760 End insert.
adamc@94 761
adam@454 762 (** 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@338 763
adam@338 764 (* begin hide *)
adam@338 765 Recursive Extraction insert.
adam@338 766 (* end hide *)
adam@283 767
adamc@94 768
adamc@86 769 (** * A Certified Regular Expression Matcher *)
adamc@86 770
adamc@93 771 (** 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.
adamc@93 772
adam@425 773 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@93 774
adamc@86 775 Require Import Ascii String.
adamc@86 776 Open Scope string_scope.
adamc@86 777
adamc@91 778 Section star.
adamc@91 779 Variable P : string -> Prop.
adamc@91 780
adamc@91 781 Inductive star : string -> Prop :=
adamc@91 782 | Empty : star ""
adamc@91 783 | Iter : forall s1 s2,
adamc@91 784 P s1
adamc@91 785 -> star s2
adamc@91 786 -> star (s1 ++ s2).
adamc@91 787 End star.
adamc@91 788
adam@283 789 (** Now we can make our first attempt at defining a [regexp] type that is indexed by predicates on strings. 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@93 790 [[
adamc@93 791 Inductive regexp : (string -> Prop) -> Set :=
adamc@93 792 | Char : forall ch : ascii,
adamc@93 793 regexp (fun s => s = String ch "")
adamc@93 794 | Concat : forall (P1 P2 : string -> Prop) (r1 : regexp P1) (r2 : regexp P2),
adamc@93 795 regexp (fun s => exists s1, exists s2, s = s1 ++ s2 /\ P1 s1 /\ P2 s2).
adamc@93 796 ]]
adamc@93 797
adam@338 798 <<
adam@338 799 User error: Large non-propositional inductive types must be in Type
adam@338 800 >>
adam@338 801
adam@454 802 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@93 803
adamc@93 804 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@93 805
adamc@89 806 Inductive regexp : (string -> Prop) -> Type :=
adamc@86 807 | Char : forall ch : ascii,
adamc@86 808 regexp (fun s => s = String ch "")
adamc@86 809 | Concat : forall P1 P2 (r1 : regexp P1) (r2 : regexp P2),
adamc@87 810 regexp (fun s => exists s1, exists s2, s = s1 ++ s2 /\ P1 s1 /\ P2 s2)
adamc@87 811 | Or : forall P1 P2 (r1 : regexp P1) (r2 : regexp P2),
adamc@91 812 regexp (fun s => P1 s \/ P2 s)
adamc@91 813 | Star : forall P (r : regexp P),
adamc@91 814 regexp (star P).
adamc@86 815
adam@425 816 (** 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@93 817
adamc@93 818 (* begin hide *)
adamc@86 819 Open Scope specif_scope.
adamc@86 820
adamc@86 821 Lemma length_emp : length "" <= 0.
adamc@86 822 crush.
adamc@86 823 Qed.
adamc@86 824
adamc@86 825 Lemma append_emp : forall s, s = "" ++ s.
adamc@86 826 crush.
adamc@86 827 Qed.
adamc@86 828
adamc@86 829 Ltac substring :=
adamc@86 830 crush;
adamc@86 831 repeat match goal with
adamc@86 832 | [ |- context[match ?N with O => _ | S _ => _ end] ] => destruct N; crush
adamc@86 833 end.
adamc@86 834
adamc@86 835 Lemma substring_le : forall s n m,
adamc@86 836 length (substring n m s) <= m.
adamc@86 837 induction s; substring.
adamc@86 838 Qed.
adamc@86 839
adamc@86 840 Lemma substring_all : forall s,
adamc@86 841 substring 0 (length s) s = s.
adamc@86 842 induction s; substring.
adamc@86 843 Qed.
adamc@86 844
adamc@86 845 Lemma substring_none : forall s n,
adamc@93 846 substring n 0 s = "".
adamc@86 847 induction s; substring.
adamc@86 848 Qed.
adamc@86 849
adam@375 850 Hint Rewrite substring_all substring_none.
adamc@86 851
adamc@86 852 Lemma substring_split : forall s m,
adamc@86 853 substring 0 m s ++ substring m (length s - m) s = s.
adamc@86 854 induction s; substring.
adamc@86 855 Qed.
adamc@86 856
adamc@86 857 Lemma length_app1 : forall s1 s2,
adamc@86 858 length s1 <= length (s1 ++ s2).
adamc@86 859 induction s1; crush.
adamc@86 860 Qed.
adamc@86 861
adamc@86 862 Hint Resolve length_emp append_emp substring_le substring_split length_app1.
adamc@86 863
adamc@86 864 Lemma substring_app_fst : forall s2 s1 n,
adamc@86 865 length s1 = n
adamc@86 866 -> substring 0 n (s1 ++ s2) = s1.
adamc@86 867 induction s1; crush.
adamc@86 868 Qed.
adamc@86 869
adamc@86 870 Lemma substring_app_snd : forall s2 s1 n,
adamc@86 871 length s1 = n
adamc@86 872 -> substring n (length (s1 ++ s2) - n) (s1 ++ s2) = s2.
adam@375 873 Hint Rewrite <- minus_n_O.
adamc@86 874
adamc@86 875 induction s1; crush.
adamc@86 876 Qed.
adamc@86 877
adam@375 878 Hint Rewrite substring_app_fst substring_app_snd using solve [trivial].
adamc@93 879 (* end hide *)
adamc@93 880
adamc@93 881 (** A few auxiliary functions help us in our final matcher definition. The function [split] will be used to implement the regexp concatenation case. *)
adamc@86 882
adamc@86 883 Section split.
adamc@86 884 Variables P1 P2 : string -> Prop.
adamc@214 885 Variable P1_dec : forall s, {P1 s} + {~ P1 s}.
adamc@214 886 Variable P2_dec : forall s, {P2 s} + {~ P2 s}.
adamc@93 887 (** We require a choice of two arbitrary string predicates and functions for deciding them. *)
adamc@86 888
adamc@86 889 Variable s : string.
adamc@93 890 (** 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. *)
adamc@93 891
adam@338 892 (** 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. *)
adamc@86 893
adam@297 894 Definition split' : forall n : nat, n <= length s
adamc@86 895 -> {exists s1, exists s2, length s1 <= n /\ s1 ++ s2 = s /\ P1 s1 /\ P2 s2}
adamc@214 896 + {forall s1 s2, length s1 <= n -> s1 ++ s2 = s -> ~ P1 s1 \/ ~ P2 s2}.
adamc@86 897 refine (fix F (n : nat) : n <= length s
adamc@86 898 -> {exists s1, exists s2, length s1 <= n /\ s1 ++ s2 = s /\ P1 s1 /\ P2 s2}
adamc@214 899 + {forall s1 s2, length s1 <= n -> s1 ++ s2 = s -> ~ P1 s1 \/ ~ P2 s2} :=
adamc@214 900 match n with
adamc@86 901 | O => fun _ => Reduce (P1_dec "" && P2_dec s)
adamc@93 902 | S n' => fun _ => (P1_dec (substring 0 (S n') s)
adamc@93 903 && P2_dec (substring (S n') (length s - S n') s))
adamc@86 904 || F n' _
adamc@86 905 end); clear F; crush; eauto 7;
adamc@86 906 match goal with
adamc@86 907 | [ _ : length ?S <= 0 |- _ ] => destruct S
adam@338 908 | [ _ : length ?S' <= S ?N |- _ ] => destruct (eq_nat_dec (length S') (S N))
adamc@86 909 end; crush.
adamc@86 910 Defined.
adamc@86 911
adam@338 912 (** 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@93 913 [[
adamc@93 914 | S n' => fun _ => let n := S n' in
adamc@93 915 (P1_dec (substring 0 n s)
adamc@93 916 && P2_dec (substring n (length s - n) s))
adamc@93 917 || F n' _
adamc@93 918 ]]
adamc@93 919
adam@338 920 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@93 921
adamc@86 922 Definition split : {exists s1, exists s2, s = s1 ++ s2 /\ P1 s1 /\ P2 s2}
adamc@214 923 + {forall s1 s2, s = s1 ++ s2 -> ~ P1 s1 \/ ~ P2 s2}.
adamc@86 924 refine (Reduce (split' (n := length s) _)); crush; eauto.
adamc@86 925 Defined.
adamc@86 926 End split.
adamc@86 927
adamc@86 928 Implicit Arguments split [P1 P2].
adamc@86 929
adamc@93 930 (* begin hide *)
adamc@91 931 Lemma app_empty_end : forall s, s ++ "" = s.
adamc@91 932 induction s; crush.
adamc@91 933 Qed.
adamc@91 934
adam@375 935 Hint Rewrite app_empty_end.
adamc@91 936
adamc@91 937 Lemma substring_self : forall s n,
adamc@91 938 n <= 0
adamc@91 939 -> substring n (length s - n) s = s.
adamc@91 940 induction s; substring.
adamc@91 941 Qed.
adamc@91 942
adamc@91 943 Lemma substring_empty : forall s n m,
adamc@91 944 m <= 0
adamc@91 945 -> substring n m s = "".
adamc@91 946 induction s; substring.
adamc@91 947 Qed.
adamc@91 948
adam@375 949 Hint Rewrite substring_self substring_empty using omega.
adamc@91 950
adamc@91 951 Lemma substring_split' : forall s n m,
adamc@91 952 substring n m s ++ substring (n + m) (length s - (n + m)) s
adamc@91 953 = substring n (length s - n) s.
adam@375 954 Hint Rewrite substring_split.
adamc@91 955
adamc@91 956 induction s; substring.
adamc@91 957 Qed.
adamc@91 958
adamc@91 959 Lemma substring_stack : forall s n2 m1 m2,
adamc@91 960 m1 <= m2
adamc@91 961 -> substring 0 m1 (substring n2 m2 s)
adamc@91 962 = substring n2 m1 s.
adamc@91 963 induction s; substring.
adamc@91 964 Qed.
adamc@91 965
adamc@91 966 Ltac substring' :=
adamc@91 967 crush;
adamc@91 968 repeat match goal with
adamc@91 969 | [ |- context[match ?N with O => _ | S _ => _ end] ] => case_eq N; crush
adamc@91 970 end.
adamc@91 971
adamc@91 972 Lemma substring_stack' : forall s n1 n2 m1 m2,
adamc@91 973 n1 + m1 <= m2
adamc@91 974 -> substring n1 m1 (substring n2 m2 s)
adamc@91 975 = substring (n1 + n2) m1 s.
adamc@91 976 induction s; substring';
adamc@91 977 match goal with
adamc@91 978 | [ |- substring ?N1 _ _ = substring ?N2 _ _ ] =>
adamc@91 979 replace N1 with N2; crush
adamc@91 980 end.
adamc@91 981 Qed.
adamc@91 982
adamc@91 983 Lemma substring_suffix : forall s n,
adamc@91 984 n <= length s
adamc@91 985 -> length (substring n (length s - n) s) = length s - n.
adamc@91 986 induction s; substring.
adamc@91 987 Qed.
adamc@91 988
adamc@91 989 Lemma substring_suffix_emp' : forall s n m,
adamc@91 990 substring n (S m) s = ""
adamc@91 991 -> n >= length s.
adamc@91 992 induction s; crush;
adamc@91 993 match goal with
adamc@91 994 | [ |- ?N >= _ ] => destruct N; crush
adamc@91 995 end;
adamc@91 996 match goal with
adamc@91 997 [ |- S ?N >= S ?E ] => assert (N >= E); [ eauto | omega ]
adamc@91 998 end.
adamc@91 999 Qed.
adamc@91 1000
adamc@91 1001 Lemma substring_suffix_emp : forall s n m,
adamc@92 1002 substring n m s = ""
adamc@92 1003 -> m > 0
adamc@91 1004 -> n >= length s.
adam@335 1005 destruct m as [ | m]; [crush | intros; apply substring_suffix_emp' with m; assumption].
adamc@91 1006 Qed.
adamc@91 1007
adamc@91 1008 Hint Rewrite substring_stack substring_stack' substring_suffix
adam@375 1009 using omega.
adamc@91 1010
adamc@91 1011 Lemma minus_minus : forall n m1 m2,
adamc@91 1012 m1 + m2 <= n
adamc@91 1013 -> n - m1 - m2 = n - (m1 + m2).
adamc@91 1014 intros; omega.
adamc@91 1015 Qed.
adamc@91 1016
adamc@91 1017 Lemma plus_n_Sm' : forall n m : nat, S (n + m) = m + S n.
adamc@91 1018 intros; omega.
adamc@91 1019 Qed.
adamc@91 1020
adam@375 1021 Hint Rewrite minus_minus using omega.
adamc@93 1022 (* end hide *)
adamc@93 1023
adamc@93 1024 (** 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 1025
adamc@91 1026 Section dec_star.
adamc@91 1027 Variable P : string -> Prop.
adamc@214 1028 Variable P_dec : forall s, {P s} + {~ P s}.
adamc@91 1029
adam@338 1030 (** 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 1031
adamc@93 1032 (* begin hide *)
adamc@91 1033 Hint Constructors star.
adamc@91 1034
adamc@91 1035 Lemma star_empty : forall s,
adamc@91 1036 length s = 0
adamc@91 1037 -> star P s.
adamc@91 1038 destruct s; crush.
adamc@91 1039 Qed.
adamc@91 1040
adamc@91 1041 Lemma star_singleton : forall s, P s -> star P s.
adamc@91 1042 intros; rewrite <- (app_empty_end s); auto.
adamc@91 1043 Qed.
adamc@91 1044
adamc@91 1045 Lemma star_app : forall s n m,
adamc@91 1046 P (substring n m s)
adamc@91 1047 -> star P (substring (n + m) (length s - (n + m)) s)
adamc@91 1048 -> star P (substring n (length s - n) s).
adamc@91 1049 induction n; substring;
adamc@91 1050 match goal with
adamc@91 1051 | [ H : P (substring ?N ?M ?S) |- _ ] =>
adamc@91 1052 solve [ rewrite <- (substring_split S M); auto
adamc@91 1053 | rewrite <- (substring_split' S N M); auto ]
adamc@91 1054 end.
adamc@91 1055 Qed.
adamc@91 1056
adamc@91 1057 Hint Resolve star_empty star_singleton star_app.
adamc@91 1058
adamc@91 1059 Variable s : string.
adamc@91 1060
adamc@91 1061 Lemma star_inv : forall s,
adamc@91 1062 star P s
adamc@91 1063 -> s = ""
adamc@91 1064 \/ exists i, i < length s
adamc@91 1065 /\ P (substring 0 (S i) s)
adamc@91 1066 /\ star P (substring (S i) (length s - S i) s).
adamc@91 1067 Hint Extern 1 (exists i : nat, _) =>
adamc@91 1068 match goal with
adamc@91 1069 | [ H : P (String _ ?S) |- _ ] => exists (length S); crush
adamc@91 1070 end.
adamc@91 1071
adamc@91 1072 induction 1; [
adamc@91 1073 crush
adamc@91 1074 | match goal with
adamc@91 1075 | [ _ : P ?S |- _ ] => destruct S; crush
adamc@91 1076 end
adamc@91 1077 ].
adamc@91 1078 Qed.
adamc@91 1079
adamc@91 1080 Lemma star_substring_inv : forall n,
adamc@91 1081 n <= length s
adamc@91 1082 -> star P (substring n (length s - n) s)
adamc@91 1083 -> substring n (length s - n) s = ""
adamc@91 1084 \/ exists l, l < length s - n
adamc@91 1085 /\ P (substring n (S l) s)
adamc@91 1086 /\ star P (substring (n + S l) (length s - (n + S l)) s).
adam@375 1087 Hint Rewrite plus_n_Sm'.
adamc@91 1088
adamc@91 1089 intros;
adamc@91 1090 match goal with
adamc@91 1091 | [ H : star _ _ |- _ ] => generalize (star_inv H); do 3 crush; eauto
adamc@91 1092 end.
adamc@91 1093 Qed.
adamc@93 1094 (* end hide *)
adamc@93 1095
adamc@93 1096 (** 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 1097
adamc@91 1098 Section dec_star''.
adamc@91 1099 Variable n : nat.
adam@454 1100 (** Variable [n] is the length of the prefix of [s] that we have already processed. *)
adamc@91 1101
adamc@91 1102 Variable P' : string -> Prop.
adamc@91 1103 Variable P'_dec : forall n' : nat, n' > n
adamc@91 1104 -> {P' (substring n' (length s - n') s)}
adamc@214 1105 + {~ P' (substring n' (length s - n') s)}.
adamc@93 1106 (** 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 1107
adamc@93 1108 (** 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']. *)
adamc@91 1109
adam@297 1110 Definition dec_star'' : forall l : nat,
adam@297 1111 {exists l', S l' <= l
adamc@91 1112 /\ P (substring n (S l') s) /\ P' (substring (n + S l') (length s - (n + S l')) s)}
adamc@91 1113 + {forall l', S l' <= l
adamc@214 1114 -> ~ P (substring n (S l') s)
adamc@214 1115 \/ ~ P' (substring (n + S l') (length s - (n + S l')) s)}.
adamc@91 1116 refine (fix F (l : nat) : {exists l', S l' <= l
adamc@91 1117 /\ P (substring n (S l') s) /\ P' (substring (n + S l') (length s - (n + S l')) s)}
adamc@91 1118 + {forall l', S l' <= l
adamc@214 1119 -> ~ P (substring n (S l') s)
adamc@214 1120 \/ ~ P' (substring (n + S l') (length s - (n + S l')) s)} :=
adam@380 1121 match l return {exists l', S l' <= l
adam@380 1122 /\ P (substring n (S l') s) /\ P' (substring (n + S l') (length s - (n + S l')) s)}
adam@380 1123 + {forall l', S l' <= l
adam@380 1124 -> ~ P (substring n (S l') s)
adam@380 1125 \/ ~ P' (substring (n + S l') (length s - (n + S l')) s)} with
adamc@91 1126 | O => _
adamc@91 1127 | S l' =>
adamc@91 1128 (P_dec (substring n (S l') s) && P'_dec (n' := n + S l') _)
adamc@91 1129 || F l'
adamc@91 1130 end); clear F; crush; eauto 7;
adamc@91 1131 match goal with
adamc@91 1132 | [ H : ?X <= S ?Y |- _ ] => destruct (eq_nat_dec X (S Y)); crush
adamc@91 1133 end.
adamc@91 1134 Defined.
adamc@91 1135 End dec_star''.
adamc@91 1136
adamc@93 1137 (* begin hide *)
adamc@92 1138 Lemma star_length_contra : forall n,
adamc@92 1139 length s > n
adamc@92 1140 -> n >= length s
adamc@92 1141 -> False.
adamc@92 1142 crush.
adamc@92 1143 Qed.
adamc@92 1144
adamc@92 1145 Lemma star_length_flip : forall n n',
adamc@92 1146 length s - n <= S n'
adamc@92 1147 -> length s > n
adamc@92 1148 -> length s - n > 0.
adamc@92 1149 crush.
adamc@92 1150 Qed.
adamc@92 1151
adamc@92 1152 Hint Resolve star_length_contra star_length_flip substring_suffix_emp.
adamc@93 1153 (* end hide *)
adamc@92 1154
adamc@93 1155 (** 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. *)
adamc@93 1156
adam@297 1157 Definition dec_star' : forall n n' : nat, length s - n' <= n
adamc@91 1158 -> {star P (substring n' (length s - n') s)}
adamc@214 1159 + {~ star P (substring n' (length s - n') s)}.
adamc@214 1160 refine (fix F (n n' : nat) : length s - n' <= n
adamc@91 1161 -> {star P (substring n' (length s - n') s)}
adamc@214 1162 + {~ star P (substring n' (length s - n') s)} :=
adamc@214 1163 match n with
adamc@91 1164 | O => fun _ => Yes
adamc@91 1165 | S n'' => fun _ =>
adamc@91 1166 le_gt_dec (length s) n'
adam@338 1167 || dec_star'' (n := n') (star P)
adam@338 1168 (fun n0 _ => Reduce (F n'' n0 _)) (length s - n')
adamc@92 1169 end); clear F; crush; eauto;
adamc@92 1170 match goal with
adamc@92 1171 | [ H : star _ _ |- _ ] => apply star_substring_inv in H; crush; eauto
adamc@92 1172 end;
adamc@92 1173 match goal with
adamc@92 1174 | [ H1 : _ < _ - _, H2 : forall l' : nat, _ <= _ - _ -> _ |- _ ] =>
adamc@92 1175 generalize (H2 _ (lt_le_S _ _ H1)); tauto
adamc@92 1176 end.
adamc@91 1177 Defined.
adamc@91 1178
adam@380 1179 (** Finally, we have [dec_star], defined by straightforward reduction from [dec_star']. *)
adamc@93 1180
adamc@214 1181 Definition dec_star : {star P s} + {~ star P s}.
adam@380 1182 refine (Reduce (dec_star' (n := length s) 0 _)); crush.
adamc@91 1183 Defined.
adamc@91 1184 End dec_star.
adamc@91 1185
adamc@93 1186 (* begin hide *)
adamc@86 1187 Lemma app_cong : forall x1 y1 x2 y2,
adamc@86 1188 x1 = x2
adamc@86 1189 -> y1 = y2
adamc@86 1190 -> x1 ++ y1 = x2 ++ y2.
adamc@86 1191 congruence.
adamc@86 1192 Qed.
adamc@86 1193
adamc@86 1194 Hint Resolve app_cong.
adamc@93 1195 (* end hide *)
adamc@93 1196
adamc@93 1197 (** 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. *)
adamc@86 1198
adam@297 1199 Definition matches : forall P (r : regexp P) s, {P s} + {~ P s}.
adamc@214 1200 refine (fix F P (r : regexp P) s : {P s} + {~ P s} :=
adamc@86 1201 match r with
adamc@86 1202 | Char ch => string_dec s (String ch "")
adamc@86 1203 | Concat _ _ r1 r2 => Reduce (split (F _ r1) (F _ r2) s)
adamc@87 1204 | Or _ _ r1 r2 => F _ r1 s || F _ r2 s
adamc@91 1205 | Star _ r => dec_star _ _ _
adamc@86 1206 end); crush;
adamc@86 1207 match goal with
adam@426 1208 | [ H : _ |- _ ] => generalize (H _ _ (eq_refl _))
adamc@93 1209 end; tauto.
adamc@86 1210 Defined.
adamc@86 1211
adam@283 1212 (** 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. *)
adam@283 1213
adamc@93 1214 (* begin hide *)
adamc@86 1215 Example hi := Concat (Char "h"%char) (Char "i"%char).
adam@380 1216 Eval hnf in matches hi "hi".
adam@380 1217 Eval hnf in matches hi "bye".
adamc@87 1218
adamc@87 1219 Example a_b := Or (Char "a"%char) (Char "b"%char).
adam@380 1220 Eval hnf in matches a_b "".
adam@380 1221 Eval hnf in matches a_b "a".
adam@380 1222 Eval hnf in matches a_b "aa".
adam@380 1223 Eval hnf in matches a_b "b".
adam@283 1224 (* end hide *)
adam@283 1225
adam@454 1226 (** Many regular expression matching problems are easy to test. The reader may run each of the following queries to verify that it gives the correct answer. We use evaluation strategy %\index{tactics!hnf}%[hnf] to reduce each term to%\index{head-normal form}% _head-normal form_, where the datatype constructor used to build its value is known. (Further reduction would involve wasteful simplification of proof terms justifying the answers of our procedures.) *)
adamc@91 1227
adamc@91 1228 Example a_star := Star (Char "a"%char).
adam@380 1229 Eval hnf in matches a_star "".
adam@380 1230 Eval hnf in matches a_star "a".
adam@380 1231 Eval hnf in matches a_star "b".
adam@380 1232 Eval hnf in matches a_star "aa".
adam@283 1233
adam@283 1234 (** Evaluation inside Coq does not scale very well, so it is easy to build other tests that run for hours or more. Such cases are better suited to execution with the extracted OCaml code. *)