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Pass through Generic, to incorporate new coqdoc features
author Adam Chlipala <adam@chlipala.net>
date Thu, 26 Jul 2012 11:22:46 -0400
parents 5f25705a10ea
children 8077352044b2
rev   line source
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 *)
adamc@84 68 Fixpoint inject (ls : list A) : ilist (length ls) :=
adamc@213 69 match ls with
adamc@84 70 | nil => Nil
adamc@84 71 | h :: t => Cons h (inject t)
adamc@84 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
adamc@213 76 Fixpoint unject n (ls : ilist n) : list A :=
adamc@84 77 match ls with
adamc@84 78 | Nil => nil
adamc@84 79 | Cons _ h t => h :: unject t
adamc@84 80 end.
adamc@84 81
adamc@84 82 Theorem inject_inverse : forall ls, unject (inject ls) = ls.
adamc@84 83 induction ls; crush.
adamc@84 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
adamc@84 91 [[
adamc@84 92 Definition hd n (ls : ilist (S n)) : A :=
adamc@84 93 match ls with
adamc@84 94 | Nil => ???
adamc@84 95 | Cons _ h _ => h
adamc@84 96 end.
adamc@213 97
adamc@213 98 ]]
adamc@84 99
adamc@84 100 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@84 101
adamc@84 102 [[
adamc@84 103 Definition hd n (ls : ilist (S n)) : A :=
adamc@84 104 match ls with
adamc@84 105 | Cons _ h _ => h
adamc@84 106 end.
adam@338 107 ]]
adamc@84 108
adam@338 109 <<
adamc@84 110 Error: Non exhaustive pattern-matching: no clause found for pattern Nil
adam@338 111 >>
adamc@84 112
adam@425 113 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@84 114
adamc@84 115 [[
adamc@84 116 Definition hd n (ls : ilist (S n)) : A :=
adamc@84 117 match ls in (ilist (S n)) with
adamc@84 118 | Cons _ h _ => h
adamc@84 119 end.
adamc@84 120 ]]
adamc@84 121
adam@338 122 <<
adam@338 123 Error: The reference n was not found in the current environment
adam@338 124 >>
adam@338 125
adam@398 126 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@84 127
adamc@84 128 Our final, working attempt at [hd] uses an auxiliary function and a surprising [return] annotation. *)
adamc@84 129
adamc@100 130 (* begin thide *)
adamc@84 131 Definition hd' n (ls : ilist n) :=
adamc@84 132 match ls in (ilist n) return (match n with O => unit | S _ => A end) with
adamc@84 133 | Nil => tt
adamc@84 134 | Cons _ h _ => h
adamc@84 135 end.
adamc@84 136
adam@283 137 Check hd'.
adam@283 138 (** %\vspace{-.15in}% [[
adam@283 139 hd'
adam@283 140 : forall n : nat, ilist n -> match n with
adam@283 141 | 0 => unit
adam@283 142 | S _ => A
adam@283 143 end
adam@283 144
adam@302 145 ]]
adam@302 146 *)
adam@283 147
adamc@84 148 Definition hd n (ls : ilist (S n)) : A := hd' ls.
adamc@100 149 (* end thide *)
adamc@84 150
adam@338 151 End ilist.
adam@338 152
adamc@84 153 (** 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@84 154
adamc@85 155
adam@371 156 (** * The One Rule of Dependent Pattern Matching in Coq *)
adam@371 157
adam@405 158 (** 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@371 159
adam@405 160 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@371 161
adam@398 162 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@371 163
adam@371 164 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@371 165 [[
adam@371 166 match E in (T x1 ... xn) as y return U with
adam@371 167 | C z1 ... zm => B
adam@371 168 | ...
adam@371 169 end
adam@371 170 ]]
adam@371 171
adam@371 172 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@371 173
adam@371 174 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@371 175
adam@371 176 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@371 177
adam@371 178 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@371 179
adam@371 180 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@371 181
adam@425 182 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@371 183
adam@371 184
adamc@85 185 (** * A Tagless Interpreter *)
adamc@85 186
adam@405 187 (** 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@85 188
adamc@85 189 Inductive type : Set :=
adamc@85 190 | Nat : type
adamc@85 191 | Bool : type
adamc@85 192 | Prod : type -> type -> type.
adamc@85 193
adamc@85 194 Inductive exp : type -> Set :=
adamc@85 195 | NConst : nat -> exp Nat
adamc@85 196 | Plus : exp Nat -> exp Nat -> exp Nat
adamc@85 197 | Eq : exp Nat -> exp Nat -> exp Bool
adamc@85 198
adamc@85 199 | BConst : bool -> exp Bool
adamc@85 200 | And : exp Bool -> exp Bool -> exp Bool
adamc@85 201 | If : forall t, exp Bool -> exp t -> exp t -> exp t
adamc@85 202
adamc@85 203 | Pair : forall t1 t2, exp t1 -> exp t2 -> exp (Prod t1 t2)
adamc@85 204 | Fst : forall t1 t2, exp (Prod t1 t2) -> exp t1
adamc@85 205 | Snd : forall t1 t2, exp (Prod t1 t2) -> exp t2.
adamc@85 206
adamc@85 207 (** 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@85 208
adam@398 209 We can give types and expressions semantics in a new style, based critically on the chance for _type-level computation_. *)
adamc@85 210
adamc@85 211 Fixpoint typeDenote (t : type) : Set :=
adamc@85 212 match t with
adamc@85 213 | Nat => nat
adamc@85 214 | Bool => bool
adamc@85 215 | Prod t1 t2 => typeDenote t1 * typeDenote t2
adamc@85 216 end%type.
adamc@85 217
adam@425 218 (** 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 identifer 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@85 219
adamc@85 220 We can define a function [expDenote] that is typed in terms of [typeDenote]. *)
adamc@85 221
adamc@213 222 Fixpoint expDenote t (e : exp t) : typeDenote t :=
adamc@213 223 match e with
adamc@85 224 | NConst n => n
adamc@85 225 | Plus e1 e2 => expDenote e1 + expDenote e2
adamc@85 226 | Eq e1 e2 => if eq_nat_dec (expDenote e1) (expDenote e2) then true else false
adamc@85 227
adamc@85 228 | BConst b => b
adamc@85 229 | And e1 e2 => expDenote e1 && expDenote e2
adamc@85 230 | If _ e' e1 e2 => if expDenote e' then expDenote e1 else expDenote e2
adamc@85 231
adamc@85 232 | Pair _ _ e1 e2 => (expDenote e1, expDenote e2)
adamc@85 233 | Fst _ _ e' => fst (expDenote e')
adamc@85 234 | Snd _ _ e' => snd (expDenote e')
adamc@85 235 end.
adamc@85 236
adamc@213 237 (** 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@85 238
adamc@85 239 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@85 240
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@417 283 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 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
adamc@85 370 We would like to do a case analysis on [cfold e1], and we attempt that in the way that has worked so far.
adamc@85 371
adamc@85 372 [[
adamc@85 373 destruct (cfold e1).
adam@338 374 ]]
adamc@85 375
adam@338 376 <<
adamc@85 377 User error: e1 is used in hypothesis e
adam@338 378 >>
adamc@85 379
adamc@85 380 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@85 381
adam@350 382 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@85 383
adamc@85 384 dep_destruct (cfold e1).
adamc@85 385
adamc@85 386 (** 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@85 387
adam@405 388 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@85 389
adamc@85 390 Restart.
adamc@85 391
adamc@85 392 induction e; crush;
adamc@85 393 repeat (match goal with
adam@405 394 | [ |- context[match cfold ?E with NConst _ => _ | _ => _ end] ] =>
adamc@213 395 dep_destruct (cfold E)
adamc@213 396 | [ |- context[match pairOut (cfold ?E) with Some _ => _
adamc@213 397 | None => _ end] ] =>
adamc@213 398 dep_destruct (cfold E)
adamc@85 399 | [ |- (if ?E then _ else _) = _ ] => destruct E
adamc@85 400 end; crush).
adamc@85 401 Qed.
adamc@100 402 (* end thide *)
adamc@86 403
adam@405 404 (** With this example, we get a first taste of how to build automated proofs that adapt automatically to changes in function definitions. *)
adam@405 405
adamc@86 406
adam@338 407 (** * Dependently Typed Red-Black Trees *)
adamc@94 408
adam@338 409 (** 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@100 410
adamc@94 411 Inductive color : Set := Red | Black.
adamc@94 412
adamc@94 413 Inductive rbtree : color -> nat -> Set :=
adamc@94 414 | Leaf : rbtree Black 0
adamc@214 415 | RedNode : forall n, rbtree Black n -> nat -> rbtree Black n -> rbtree Red n
adamc@94 416 | BlackNode : forall c1 c2 n, rbtree c1 n -> nat -> rbtree c2 n -> rbtree Black (S n).
adamc@94 417
adamc@214 418 (** 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@214 419
adamc@214 420 (** 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@214 421
adamc@100 422 (* EX: Prove that every [rbtree] is balanced. *)
adamc@100 423
adamc@100 424 (* begin thide *)
adamc@95 425 Require Import Max Min.
adamc@95 426
adamc@95 427 Section depth.
adamc@95 428 Variable f : nat -> nat -> nat.
adamc@95 429
adamc@214 430 Fixpoint depth c n (t : rbtree c n) : nat :=
adamc@95 431 match t with
adamc@95 432 | Leaf => 0
adamc@95 433 | RedNode _ t1 _ t2 => S (f (depth t1) (depth t2))
adamc@95 434 | BlackNode _ _ _ t1 _ t2 => S (f (depth t1) (depth t2))
adamc@95 435 end.
adamc@95 436 End depth.
adamc@95 437
adam@338 438 (** 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@214 439
adam@283 440 Check min_dec.
adam@283 441 (** %\vspace{-.15in}% [[
adam@283 442 min_dec
adam@283 443 : forall n m : nat, {min n m = n} + {min n m = m}
adam@302 444 ]]
adam@302 445 *)
adam@283 446
adamc@95 447 Theorem depth_min : forall c n (t : rbtree c n), depth min t >= n.
adamc@95 448 induction t; crush;
adamc@95 449 match goal with
adamc@95 450 | [ |- context[min ?X ?Y] ] => destruct (min_dec X Y)
adamc@95 451 end; crush.
adamc@95 452 Qed.
adamc@95 453
adamc@214 454 (** There is an analogous upper-bound theorem based on black depth. Unfortunately, a symmetric proof script does not suffice to establish it. *)
adamc@214 455
adamc@214 456 Theorem depth_max : forall c n (t : rbtree c n), depth max t <= 2 * n + 1.
adamc@214 457 induction t; crush;
adamc@214 458 match goal with
adamc@214 459 | [ |- context[max ?X ?Y] ] => destruct (max_dec X Y)
adamc@214 460 end; crush.
adamc@214 461
adamc@214 462 (** Two subgoals remain. One of them is: [[
adamc@214 463 n : nat
adamc@214 464 t1 : rbtree Black n
adamc@214 465 n0 : nat
adamc@214 466 t2 : rbtree Black n
adamc@214 467 IHt1 : depth max t1 <= n + (n + 0) + 1
adamc@214 468 IHt2 : depth max t2 <= n + (n + 0) + 1
adamc@214 469 e : max (depth max t1) (depth max t2) = depth max t1
adamc@214 470 ============================
adamc@214 471 S (depth max t1) <= n + (n + 0) + 1
adamc@214 472
adamc@214 473 ]]
adamc@214 474
adam@398 475 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@214 476
adamc@214 477 Abort.
adamc@214 478
adamc@214 479 (** 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@214 480
adamc@95 481 Lemma depth_max' : forall c n (t : rbtree c n), match c with
adamc@95 482 | Red => depth max t <= 2 * n + 1
adamc@95 483 | Black => depth max t <= 2 * n
adamc@95 484 end.
adamc@95 485 induction t; crush;
adamc@95 486 match goal with
adamc@95 487 | [ |- context[max ?X ?Y] ] => destruct (max_dec X Y)
adamc@100 488 end; crush;
adamc@100 489 repeat (match goal with
adamc@214 490 | [ H : context[match ?C with Red => _ | Black => _ end] |- _ ] =>
adamc@214 491 destruct C
adamc@100 492 end; crush).
adamc@95 493 Qed.
adamc@95 494
adam@338 495 (** 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@214 496
adamc@95 497 Theorem depth_max : forall c n (t : rbtree c n), depth max t <= 2 * n + 1.
adamc@95 498 intros; generalize (depth_max' t); destruct c; crush.
adamc@95 499 Qed.
adamc@95 500
adamc@214 501 (** 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@214 502
adamc@95 503 Theorem balanced : forall c n (t : rbtree c n), 2 * depth min t + 1 >= depth max t.
adamc@95 504 intros; generalize (depth_min t); generalize (depth_max t); crush.
adamc@95 505 Qed.
adamc@100 506 (* end thide *)
adamc@95 507
adamc@214 508 (** 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@95 509
adamc@94 510 Inductive rtree : nat -> Set :=
adamc@94 511 | RedNode' : forall c1 c2 n, rbtree c1 n -> nat -> rbtree c2 n -> rtree n.
adamc@94 512
adam@338 513 (** 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@214 514
adamc@96 515 Section present.
adamc@96 516 Variable x : nat.
adamc@96 517
adamc@214 518 Fixpoint present c n (t : rbtree c n) : Prop :=
adamc@96 519 match t with
adamc@96 520 | Leaf => False
adamc@96 521 | RedNode _ a y b => present a \/ x = y \/ present b
adamc@96 522 | BlackNode _ _ _ a y b => present a \/ x = y \/ present b
adamc@96 523 end.
adamc@96 524
adamc@96 525 Definition rpresent n (t : rtree n) : Prop :=
adamc@96 526 match t with
adamc@96 527 | RedNode' _ _ _ a y b => present a \/ x = y \/ present b
adamc@96 528 end.
adamc@96 529 End present.
adamc@96 530
adam@425 531 (* begin hide *)
adam@425 532 Definition sigT' := sigT.
adam@425 533 (* end hide *)
adam@425 534
adam@338 535 (** 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@214 536
adamc@100 537 Locate "{ _ : _ & _ }".
adamc@214 538 (** [[
adamc@214 539 Notation Scope
adamc@214 540 "{ x : A & P }" := sigT (fun x : A => P)
adam@302 541 ]]
adam@302 542 *)
adamc@214 543
adamc@100 544 Print sigT.
adamc@214 545 (** [[
adamc@214 546 Inductive sigT (A : Type) (P : A -> Type) : Type :=
adamc@214 547 existT : forall x : A, P x -> sigT P
adam@302 548 ]]
adam@302 549 *)
adamc@214 550
adamc@214 551 (** It will be helpful to define a concise notation for the constructor of [sigT]. *)
adamc@100 552
adamc@94 553 Notation "{< x >}" := (existT _ _ x).
adamc@94 554
adamc@214 555 (** 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@214 556
adam@338 557 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@338 558
adam@425 559 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@214 560
adamc@94 561 Definition balance1 n (a : rtree n) (data : nat) c2 :=
adamc@214 562 match a in rtree n return rbtree c2 n
adamc@214 563 -> { c : color & rbtree c (S n) } with
adam@380 564 | RedNode' _ c0 _ t1 y t2 =>
adam@380 565 match t1 in rbtree c n return rbtree c0 n -> rbtree c2 n
adamc@214 566 -> { c : color & rbtree c (S n) } with
adamc@214 567 | RedNode _ a x b => fun c d =>
adamc@214 568 {<RedNode (BlackNode a x b) y (BlackNode c data d)>}
adamc@94 569 | t1' => fun t2 =>
adam@380 570 match t2 in rbtree c n return rbtree Black n -> rbtree c2 n
adamc@214 571 -> { c : color & rbtree c (S n) } with
adamc@214 572 | RedNode _ b x c => fun a d =>
adamc@214 573 {<RedNode (BlackNode a y b) x (BlackNode c data d)>}
adamc@95 574 | b => fun a t => {<BlackNode (RedNode a y b) data t>}
adamc@94 575 end t1'
adamc@94 576 end t2
adamc@94 577 end.
adamc@94 578
adam@405 579 (** 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@214 580
adam@425 581 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@214 582
adam@338 583 Here is the symmetric function [balance2], for cases where the possibly invalid tree appears on the right rather than on the left. *)
adamc@214 584
adamc@94 585 Definition balance2 n (a : rtree n) (data : nat) c2 :=
adamc@94 586 match a in rtree n return rbtree c2 n -> { c : color & rbtree c (S n) } with
adam@380 587 | RedNode' _ c0 _ t1 z t2 =>
adam@380 588 match t1 in rbtree c n return rbtree c0 n -> rbtree c2 n
adamc@214 589 -> { c : color & rbtree c (S n) } with
adamc@214 590 | RedNode _ b y c => fun d a =>
adamc@214 591 {<RedNode (BlackNode a data b) y (BlackNode c z d)>}
adamc@94 592 | t1' => fun t2 =>
adam@380 593 match t2 in rbtree c n return rbtree Black n -> rbtree c2 n
adamc@214 594 -> { c : color & rbtree c (S n) } with
adamc@214 595 | RedNode _ c z' d => fun b a =>
adamc@214 596 {<RedNode (BlackNode a data b) z (BlackNode c z' d)>}
adamc@95 597 | b => fun a t => {<BlackNode t data (RedNode a z b)>}
adamc@94 598 end t1'
adamc@94 599 end t2
adamc@94 600 end.
adamc@94 601
adamc@214 602 (** 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@214 603
adamc@94 604 Section insert.
adamc@94 605 Variable x : nat.
adamc@94 606
adamc@214 607 (** 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@214 608
adamc@94 609 Definition insResult c n :=
adamc@94 610 match c with
adamc@94 611 | Red => rtree n
adamc@94 612 | Black => { c' : color & rbtree c' n }
adamc@94 613 end.
adamc@94 614
adam@338 615 (** 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@214 616
adamc@214 617 Here is the definition of [ins]. Again, we do not want to dwell on the functional details. *)
adamc@214 618
adamc@214 619 Fixpoint ins c n (t : rbtree c n) : insResult c n :=
adamc@214 620 match t with
adamc@94 621 | Leaf => {< RedNode Leaf x Leaf >}
adamc@94 622 | RedNode _ a y b =>
adamc@94 623 if le_lt_dec x y
adamc@94 624 then RedNode' (projT2 (ins a)) y b
adamc@94 625 else RedNode' a y (projT2 (ins b))
adamc@94 626 | BlackNode c1 c2 _ a y b =>
adamc@94 627 if le_lt_dec x y
adamc@94 628 then
adamc@94 629 match c1 return insResult c1 _ -> _ with
adamc@94 630 | Red => fun ins_a => balance1 ins_a y b
adamc@94 631 | _ => fun ins_a => {< BlackNode (projT2 ins_a) y b >}
adamc@94 632 end (ins a)
adamc@94 633 else
adamc@94 634 match c2 return insResult c2 _ -> _ with
adamc@94 635 | Red => fun ins_b => balance2 ins_b y a
adamc@94 636 | _ => fun ins_b => {< BlackNode a y (projT2 ins_b) >}
adamc@94 637 end (ins b)
adamc@94 638 end.
adamc@94 639
adam@398 640 (** 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@214 641
adamc@214 642 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@214 643
adamc@94 644 Definition insertResult c n :=
adamc@94 645 match c with
adamc@94 646 | Red => rbtree Black (S n)
adamc@94 647 | Black => { c' : color & rbtree c' n }
adamc@94 648 end.
adamc@94 649
adamc@214 650 (** A simple clean-up procedure translates [insResult]s into [insertResult]s. *)
adamc@214 651
adamc@97 652 Definition makeRbtree c n : insResult c n -> insertResult c n :=
adamc@214 653 match c with
adamc@94 654 | Red => fun r =>
adamc@214 655 match r with
adamc@94 656 | RedNode' _ _ _ a x b => BlackNode a x b
adamc@94 657 end
adamc@94 658 | Black => fun r => r
adamc@94 659 end.
adamc@94 660
adamc@214 661 (** 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@214 662
adamc@97 663 Implicit Arguments makeRbtree [c n].
adamc@94 664
adamc@214 665 (** Finally, we define [insert] as a simple composition of [ins] and [makeRbtree]. *)
adamc@214 666
adamc@94 667 Definition insert c n (t : rbtree c n) : insertResult c n :=
adamc@97 668 makeRbtree (ins t).
adamc@94 669
adamc@214 670 (** 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@214 671
adamc@95 672 Section present.
adamc@95 673 Variable z : nat.
adamc@95 674
adamc@214 675 (** 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@214 676
adam@367 677 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@214 678
adamc@98 679 Ltac present_balance :=
adamc@98 680 crush;
adamc@98 681 repeat (match goal with
adam@425 682 | [ _ : context[match ?T with Leaf => _ | _ => _ end] |- _ ] =>
adam@425 683 dep_destruct T
adam@405 684 | [ |- context[match ?T with Leaf => _ | _ => _ end] ] => dep_destruct T
adamc@98 685 end; crush).
adamc@98 686
adamc@214 687 (** 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@214 688
adam@294 689 Lemma present_balance1 : forall n (a : rtree n) (y : nat) c2 (b : rbtree c2 n),
adamc@95 690 present z (projT2 (balance1 a y b))
adamc@95 691 <-> rpresent z a \/ z = y \/ present z b.
adamc@98 692 destruct a; present_balance.
adamc@95 693 Qed.
adamc@95 694
adamc@213 695 Lemma present_balance2 : forall n (a : rtree n) (y : nat) c2 (b : rbtree c2 n),
adamc@95 696 present z (projT2 (balance2 a y b))
adamc@95 697 <-> rpresent z a \/ z = y \/ present z b.
adamc@98 698 destruct a; present_balance.
adamc@95 699 Qed.
adamc@95 700
adamc@214 701 (** 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@214 702
adamc@95 703 Definition present_insResult c n :=
adamc@95 704 match c return (rbtree c n -> insResult c n -> Prop) with
adamc@95 705 | Red => fun t r => rpresent z r <-> z = x \/ present z t
adamc@95 706 | Black => fun t r => present z (projT2 r) <-> z = x \/ present z t
adamc@95 707 end.
adamc@95 708
adamc@214 709 (** 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@214 710
adamc@95 711 Theorem present_ins : forall c n (t : rbtree c n),
adamc@95 712 present_insResult t (ins t).
adamc@95 713 induction t; crush;
adamc@95 714 repeat (match goal with
adam@338 715 | [ _ : context[if ?E then _ else _] |- _ ] => destruct E
adamc@95 716 | [ |- context[if ?E then _ else _] ] => destruct E
adam@338 717 | [ _ : context[match ?C with Red => _ | Black => _ end]
adamc@214 718 |- _ ] => destruct C
adamc@95 719 end; crush);
adamc@95 720 try match goal with
adam@338 721 | [ _ : context[balance1 ?A ?B ?C] |- _ ] =>
adamc@95 722 generalize (present_balance1 A B C)
adamc@95 723 end;
adamc@95 724 try match goal with
adam@338 725 | [ _ : context[balance2 ?A ?B ?C] |- _ ] =>
adamc@95 726 generalize (present_balance2 A B C)
adamc@95 727 end;
adamc@95 728 try match goal with
adamc@95 729 | [ |- context[balance1 ?A ?B ?C] ] =>
adamc@95 730 generalize (present_balance1 A B C)
adamc@95 731 end;
adamc@95 732 try match goal with
adamc@95 733 | [ |- context[balance2 ?A ?B ?C] ] =>
adamc@95 734 generalize (present_balance2 A B C)
adamc@95 735 end;
adamc@214 736 crush;
adamc@95 737 match goal with
adamc@95 738 | [ z : nat, x : nat |- _ ] =>
adamc@95 739 match goal with
adamc@95 740 | [ H : z = x |- _ ] => rewrite H in *; clear H
adamc@95 741 end
adamc@95 742 end;
adamc@95 743 tauto.
adamc@95 744 Qed.
adamc@95 745
adamc@214 746 (** 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@214 747
adamc@213 748 Ltac present_insert :=
adamc@213 749 unfold insert; intros n t; inversion t;
adamc@97 750 generalize (present_ins t); simpl;
adamc@97 751 dep_destruct (ins t); tauto.
adamc@97 752
adamc@95 753 Theorem present_insert_Red : forall n (t : rbtree Red n),
adamc@95 754 present z (insert t)
adamc@95 755 <-> (z = x \/ present z t).
adamc@213 756 present_insert.
adamc@95 757 Qed.
adamc@95 758
adamc@95 759 Theorem present_insert_Black : forall n (t : rbtree Black n),
adamc@95 760 present z (projT2 (insert t))
adamc@95 761 <-> (z = x \/ present z t).
adamc@213 762 present_insert.
adamc@95 763 Qed.
adamc@95 764 End present.
adamc@94 765 End insert.
adamc@94 766
adam@425 767 (** 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 which 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@338 768
adam@338 769 (* begin hide *)
adam@338 770 Recursive Extraction insert.
adam@338 771 (* end hide *)
adam@283 772
adamc@94 773
adamc@86 774 (** * A Certified Regular Expression Matcher *)
adamc@86 775
adamc@93 776 (** 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@93 777
adam@425 778 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@93 779
adamc@86 780 Require Import Ascii String.
adamc@86 781 Open Scope string_scope.
adamc@86 782
adamc@91 783 Section star.
adamc@91 784 Variable P : string -> Prop.
adamc@91 785
adamc@91 786 Inductive star : string -> Prop :=
adamc@91 787 | Empty : star ""
adamc@91 788 | Iter : forall s1 s2,
adamc@91 789 P s1
adamc@91 790 -> star s2
adamc@91 791 -> star (s1 ++ s2).
adamc@91 792 End star.
adamc@91 793
adam@283 794 (** 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@93 795 [[
adamc@93 796 Inductive regexp : (string -> Prop) -> Set :=
adamc@93 797 | Char : forall ch : ascii,
adamc@93 798 regexp (fun s => s = String ch "")
adamc@93 799 | Concat : forall (P1 P2 : string -> Prop) (r1 : regexp P1) (r2 : regexp P2),
adamc@93 800 regexp (fun s => exists s1, exists s2, s = s1 ++ s2 /\ P1 s1 /\ P2 s2).
adamc@93 801 ]]
adamc@93 802
adam@338 803 <<
adam@338 804 User error: Large non-propositional inductive types must be in Type
adam@338 805 >>
adam@338 806
adam@338 807 What is a %\index{large inductive types}%large inductive type? In Coq, it is an inductive type that has a constructor which 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@93 808
adamc@93 809 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@93 810
adamc@89 811 Inductive regexp : (string -> Prop) -> Type :=
adamc@86 812 | Char : forall ch : ascii,
adamc@86 813 regexp (fun s => s = String ch "")
adamc@86 814 | Concat : forall P1 P2 (r1 : regexp P1) (r2 : regexp P2),
adamc@87 815 regexp (fun s => exists s1, exists s2, s = s1 ++ s2 /\ P1 s1 /\ P2 s2)
adamc@87 816 | Or : forall P1 P2 (r1 : regexp P1) (r2 : regexp P2),
adamc@91 817 regexp (fun s => P1 s \/ P2 s)
adamc@91 818 | Star : forall P (r : regexp P),
adamc@91 819 regexp (star P).
adamc@86 820
adam@425 821 (** 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@93 822
adamc@93 823 (* begin hide *)
adamc@86 824 Open Scope specif_scope.
adamc@86 825
adamc@86 826 Lemma length_emp : length "" <= 0.
adamc@86 827 crush.
adamc@86 828 Qed.
adamc@86 829
adamc@86 830 Lemma append_emp : forall s, s = "" ++ s.
adamc@86 831 crush.
adamc@86 832 Qed.
adamc@86 833
adamc@86 834 Ltac substring :=
adamc@86 835 crush;
adamc@86 836 repeat match goal with
adamc@86 837 | [ |- context[match ?N with O => _ | S _ => _ end] ] => destruct N; crush
adamc@86 838 end.
adamc@86 839
adamc@86 840 Lemma substring_le : forall s n m,
adamc@86 841 length (substring n m s) <= m.
adamc@86 842 induction s; substring.
adamc@86 843 Qed.
adamc@86 844
adamc@86 845 Lemma substring_all : forall s,
adamc@86 846 substring 0 (length s) s = s.
adamc@86 847 induction s; substring.
adamc@86 848 Qed.
adamc@86 849
adamc@86 850 Lemma substring_none : forall s n,
adamc@93 851 substring n 0 s = "".
adamc@86 852 induction s; substring.
adamc@86 853 Qed.
adamc@86 854
adam@375 855 Hint Rewrite substring_all substring_none.
adamc@86 856
adamc@86 857 Lemma substring_split : forall s m,
adamc@86 858 substring 0 m s ++ substring m (length s - m) s = s.
adamc@86 859 induction s; substring.
adamc@86 860 Qed.
adamc@86 861
adamc@86 862 Lemma length_app1 : forall s1 s2,
adamc@86 863 length s1 <= length (s1 ++ s2).
adamc@86 864 induction s1; crush.
adamc@86 865 Qed.
adamc@86 866
adamc@86 867 Hint Resolve length_emp append_emp substring_le substring_split length_app1.
adamc@86 868
adamc@86 869 Lemma substring_app_fst : forall s2 s1 n,
adamc@86 870 length s1 = n
adamc@86 871 -> substring 0 n (s1 ++ s2) = s1.
adamc@86 872 induction s1; crush.
adamc@86 873 Qed.
adamc@86 874
adamc@86 875 Lemma substring_app_snd : forall s2 s1 n,
adamc@86 876 length s1 = n
adamc@86 877 -> substring n (length (s1 ++ s2) - n) (s1 ++ s2) = s2.
adam@375 878 Hint Rewrite <- minus_n_O.
adamc@86 879
adamc@86 880 induction s1; crush.
adamc@86 881 Qed.
adamc@86 882
adam@375 883 Hint Rewrite substring_app_fst substring_app_snd using solve [trivial].
adamc@93 884 (* end hide *)
adamc@93 885
adamc@93 886 (** A few auxiliary functions help us in our final matcher definition. The function [split] will be used to implement the regexp concatenation case. *)
adamc@86 887
adamc@86 888 Section split.
adamc@86 889 Variables P1 P2 : string -> Prop.
adamc@214 890 Variable P1_dec : forall s, {P1 s} + {~ P1 s}.
adamc@214 891 Variable P2_dec : forall s, {P2 s} + {~ P2 s}.
adamc@93 892 (** We require a choice of two arbitrary string predicates and functions for deciding them. *)
adamc@86 893
adamc@86 894 Variable s : string.
adamc@93 895 (** 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@93 896
adam@338 897 (** 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@86 898
adam@297 899 Definition split' : forall n : nat, n <= length s
adamc@86 900 -> {exists s1, exists s2, length s1 <= n /\ s1 ++ s2 = s /\ P1 s1 /\ P2 s2}
adamc@214 901 + {forall s1 s2, length s1 <= n -> s1 ++ s2 = s -> ~ P1 s1 \/ ~ P2 s2}.
adamc@86 902 refine (fix F (n : nat) : n <= length s
adamc@86 903 -> {exists s1, exists s2, length s1 <= n /\ s1 ++ s2 = s /\ P1 s1 /\ P2 s2}
adamc@214 904 + {forall s1 s2, length s1 <= n -> s1 ++ s2 = s -> ~ P1 s1 \/ ~ P2 s2} :=
adamc@214 905 match n with
adamc@86 906 | O => fun _ => Reduce (P1_dec "" && P2_dec s)
adamc@93 907 | S n' => fun _ => (P1_dec (substring 0 (S n') s)
adamc@93 908 && P2_dec (substring (S n') (length s - S n') s))
adamc@86 909 || F n' _
adamc@86 910 end); clear F; crush; eauto 7;
adamc@86 911 match goal with
adamc@86 912 | [ _ : length ?S <= 0 |- _ ] => destruct S
adam@338 913 | [ _ : length ?S' <= S ?N |- _ ] => destruct (eq_nat_dec (length S') (S N))
adamc@86 914 end; crush.
adamc@86 915 Defined.
adamc@86 916
adam@338 917 (** 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@93 918 [[
adamc@93 919 | S n' => fun _ => let n := S n' in
adamc@93 920 (P1_dec (substring 0 n s)
adamc@93 921 && P2_dec (substring n (length s - n) s))
adamc@93 922 || F n' _
adamc@214 923
adamc@93 924 ]]
adamc@93 925
adam@338 926 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@93 927
adamc@86 928 Definition split : {exists s1, exists s2, s = s1 ++ s2 /\ P1 s1 /\ P2 s2}
adamc@214 929 + {forall s1 s2, s = s1 ++ s2 -> ~ P1 s1 \/ ~ P2 s2}.
adamc@86 930 refine (Reduce (split' (n := length s) _)); crush; eauto.
adamc@86 931 Defined.
adamc@86 932 End split.
adamc@86 933
adamc@86 934 Implicit Arguments split [P1 P2].
adamc@86 935
adamc@93 936 (* begin hide *)
adamc@91 937 Lemma app_empty_end : forall s, s ++ "" = s.
adamc@91 938 induction s; crush.
adamc@91 939 Qed.
adamc@91 940
adam@375 941 Hint Rewrite app_empty_end.
adamc@91 942
adamc@91 943 Lemma substring_self : forall s n,
adamc@91 944 n <= 0
adamc@91 945 -> substring n (length s - n) s = s.
adamc@91 946 induction s; substring.
adamc@91 947 Qed.
adamc@91 948
adamc@91 949 Lemma substring_empty : forall s n m,
adamc@91 950 m <= 0
adamc@91 951 -> substring n m s = "".
adamc@91 952 induction s; substring.
adamc@91 953 Qed.
adamc@91 954
adam@375 955 Hint Rewrite substring_self substring_empty using omega.
adamc@91 956
adamc@91 957 Lemma substring_split' : forall s n m,
adamc@91 958 substring n m s ++ substring (n + m) (length s - (n + m)) s
adamc@91 959 = substring n (length s - n) s.
adam@375 960 Hint Rewrite substring_split.
adamc@91 961
adamc@91 962 induction s; substring.
adamc@91 963 Qed.
adamc@91 964
adamc@91 965 Lemma substring_stack : forall s n2 m1 m2,
adamc@91 966 m1 <= m2
adamc@91 967 -> substring 0 m1 (substring n2 m2 s)
adamc@91 968 = substring n2 m1 s.
adamc@91 969 induction s; substring.
adamc@91 970 Qed.
adamc@91 971
adamc@91 972 Ltac substring' :=
adamc@91 973 crush;
adamc@91 974 repeat match goal with
adamc@91 975 | [ |- context[match ?N with O => _ | S _ => _ end] ] => case_eq N; crush
adamc@91 976 end.
adamc@91 977
adamc@91 978 Lemma substring_stack' : forall s n1 n2 m1 m2,
adamc@91 979 n1 + m1 <= m2
adamc@91 980 -> substring n1 m1 (substring n2 m2 s)
adamc@91 981 = substring (n1 + n2) m1 s.
adamc@91 982 induction s; substring';
adamc@91 983 match goal with
adamc@91 984 | [ |- substring ?N1 _ _ = substring ?N2 _ _ ] =>
adamc@91 985 replace N1 with N2; crush
adamc@91 986 end.
adamc@91 987 Qed.
adamc@91 988
adamc@91 989 Lemma substring_suffix : forall s n,
adamc@91 990 n <= length s
adamc@91 991 -> length (substring n (length s - n) s) = length s - n.
adamc@91 992 induction s; substring.
adamc@91 993 Qed.
adamc@91 994
adamc@91 995 Lemma substring_suffix_emp' : forall s n m,
adamc@91 996 substring n (S m) s = ""
adamc@91 997 -> n >= length s.
adamc@91 998 induction s; crush;
adamc@91 999 match goal with
adamc@91 1000 | [ |- ?N >= _ ] => destruct N; crush
adamc@91 1001 end;
adamc@91 1002 match goal with
adamc@91 1003 [ |- S ?N >= S ?E ] => assert (N >= E); [ eauto | omega ]
adamc@91 1004 end.
adamc@91 1005 Qed.
adamc@91 1006
adamc@91 1007 Lemma substring_suffix_emp : forall s n m,
adamc@92 1008 substring n m s = ""
adamc@92 1009 -> m > 0
adamc@91 1010 -> n >= length s.
adam@335 1011 destruct m as [ | m]; [crush | intros; apply substring_suffix_emp' with m; assumption].
adamc@91 1012 Qed.
adamc@91 1013
adamc@91 1014 Hint Rewrite substring_stack substring_stack' substring_suffix
adam@375 1015 using omega.
adamc@91 1016
adamc@91 1017 Lemma minus_minus : forall n m1 m2,
adamc@91 1018 m1 + m2 <= n
adamc@91 1019 -> n - m1 - m2 = n - (m1 + m2).
adamc@91 1020 intros; omega.
adamc@91 1021 Qed.
adamc@91 1022
adamc@91 1023 Lemma plus_n_Sm' : forall n m : nat, S (n + m) = m + S n.
adamc@91 1024 intros; omega.
adamc@91 1025 Qed.
adamc@91 1026
adam@375 1027 Hint Rewrite minus_minus using omega.
adamc@93 1028 (* end hide *)
adamc@93 1029
adamc@93 1030 (** 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 1031
adamc@91 1032 Section dec_star.
adamc@91 1033 Variable P : string -> Prop.
adamc@214 1034 Variable P_dec : forall s, {P s} + {~ P s}.
adamc@91 1035
adam@338 1036 (** 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 1037
adamc@93 1038 (* begin hide *)
adamc@91 1039 Hint Constructors star.
adamc@91 1040
adamc@91 1041 Lemma star_empty : forall s,
adamc@91 1042 length s = 0
adamc@91 1043 -> star P s.
adamc@91 1044 destruct s; crush.
adamc@91 1045 Qed.
adamc@91 1046
adamc@91 1047 Lemma star_singleton : forall s, P s -> star P s.
adamc@91 1048 intros; rewrite <- (app_empty_end s); auto.
adamc@91 1049 Qed.
adamc@91 1050
adamc@91 1051 Lemma star_app : forall s n m,
adamc@91 1052 P (substring n m s)
adamc@91 1053 -> star P (substring (n + m) (length s - (n + m)) s)
adamc@91 1054 -> star P (substring n (length s - n) s).
adamc@91 1055 induction n; substring;
adamc@91 1056 match goal with
adamc@91 1057 | [ H : P (substring ?N ?M ?S) |- _ ] =>
adamc@91 1058 solve [ rewrite <- (substring_split S M); auto
adamc@91 1059 | rewrite <- (substring_split' S N M); auto ]
adamc@91 1060 end.
adamc@91 1061 Qed.
adamc@91 1062
adamc@91 1063 Hint Resolve star_empty star_singleton star_app.
adamc@91 1064
adamc@91 1065 Variable s : string.
adamc@91 1066
adamc@91 1067 Lemma star_inv : forall s,
adamc@91 1068 star P s
adamc@91 1069 -> s = ""
adamc@91 1070 \/ exists i, i < length s
adamc@91 1071 /\ P (substring 0 (S i) s)
adamc@91 1072 /\ star P (substring (S i) (length s - S i) s).
adamc@91 1073 Hint Extern 1 (exists i : nat, _) =>
adamc@91 1074 match goal with
adamc@91 1075 | [ H : P (String _ ?S) |- _ ] => exists (length S); crush
adamc@91 1076 end.
adamc@91 1077
adamc@91 1078 induction 1; [
adamc@91 1079 crush
adamc@91 1080 | match goal with
adamc@91 1081 | [ _ : P ?S |- _ ] => destruct S; crush
adamc@91 1082 end
adamc@91 1083 ].
adamc@91 1084 Qed.
adamc@91 1085
adamc@91 1086 Lemma star_substring_inv : forall n,
adamc@91 1087 n <= length s
adamc@91 1088 -> star P (substring n (length s - n) s)
adamc@91 1089 -> substring n (length s - n) s = ""
adamc@91 1090 \/ exists l, l < length s - n
adamc@91 1091 /\ P (substring n (S l) s)
adamc@91 1092 /\ star P (substring (n + S l) (length s - (n + S l)) s).
adam@375 1093 Hint Rewrite plus_n_Sm'.
adamc@91 1094
adamc@91 1095 intros;
adamc@91 1096 match goal with
adamc@91 1097 | [ H : star _ _ |- _ ] => generalize (star_inv H); do 3 crush; eauto
adamc@91 1098 end.
adamc@91 1099 Qed.
adamc@93 1100 (* end hide *)
adamc@93 1101
adamc@93 1102 (** 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 1103
adamc@91 1104 Section dec_star''.
adamc@91 1105 Variable n : nat.
adamc@93 1106 (** [n] is the length of the prefix of [s] that we have already processed. *)
adamc@91 1107
adamc@91 1108 Variable P' : string -> Prop.
adamc@91 1109 Variable P'_dec : forall n' : nat, n' > n
adamc@91 1110 -> {P' (substring n' (length s - n') s)}
adamc@214 1111 + {~ P' (substring n' (length s - n') s)}.
adamc@93 1112 (** 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 1113
adamc@93 1114 (** 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 1115
adam@297 1116 Definition dec_star'' : forall l : nat,
adam@297 1117 {exists l', S l' <= l
adamc@91 1118 /\ P (substring n (S l') s) /\ P' (substring (n + S l') (length s - (n + S l')) s)}
adamc@91 1119 + {forall l', S l' <= l
adamc@214 1120 -> ~ P (substring n (S l') s)
adamc@214 1121 \/ ~ P' (substring (n + S l') (length s - (n + S l')) s)}.
adamc@91 1122 refine (fix F (l : nat) : {exists l', S l' <= l
adamc@91 1123 /\ P (substring n (S l') s) /\ P' (substring (n + S l') (length s - (n + S l')) s)}
adamc@91 1124 + {forall l', S l' <= l
adamc@214 1125 -> ~ P (substring n (S l') s)
adamc@214 1126 \/ ~ P' (substring (n + S l') (length s - (n + S l')) s)} :=
adam@380 1127 match l return {exists l', S l' <= l
adam@380 1128 /\ P (substring n (S l') s) /\ P' (substring (n + S l') (length s - (n + S l')) s)}
adam@380 1129 + {forall l', S l' <= l
adam@380 1130 -> ~ P (substring n (S l') s)
adam@380 1131 \/ ~ P' (substring (n + S l') (length s - (n + S l')) s)} with
adamc@91 1132 | O => _
adamc@91 1133 | S l' =>
adamc@91 1134 (P_dec (substring n (S l') s) && P'_dec (n' := n + S l') _)
adamc@91 1135 || F l'
adamc@91 1136 end); clear F; crush; eauto 7;
adamc@91 1137 match goal with
adamc@91 1138 | [ H : ?X <= S ?Y |- _ ] => destruct (eq_nat_dec X (S Y)); crush
adamc@91 1139 end.
adamc@91 1140 Defined.
adamc@91 1141 End dec_star''.
adamc@91 1142
adamc@93 1143 (* begin hide *)
adamc@92 1144 Lemma star_length_contra : forall n,
adamc@92 1145 length s > n
adamc@92 1146 -> n >= length s
adamc@92 1147 -> False.
adamc@92 1148 crush.
adamc@92 1149 Qed.
adamc@92 1150
adamc@92 1151 Lemma star_length_flip : forall n n',
adamc@92 1152 length s - n <= S n'
adamc@92 1153 -> length s > n
adamc@92 1154 -> length s - n > 0.
adamc@92 1155 crush.
adamc@92 1156 Qed.
adamc@92 1157
adamc@92 1158 Hint Resolve star_length_contra star_length_flip substring_suffix_emp.
adamc@93 1159 (* end hide *)
adamc@92 1160
adamc@93 1161 (** 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 1162
adam@297 1163 Definition dec_star' : forall n n' : nat, length s - n' <= n
adamc@91 1164 -> {star P (substring n' (length s - n') s)}
adamc@214 1165 + {~ star P (substring n' (length s - n') s)}.
adamc@214 1166 refine (fix F (n n' : nat) : length s - n' <= n
adamc@91 1167 -> {star P (substring n' (length s - n') s)}
adamc@214 1168 + {~ star P (substring n' (length s - n') s)} :=
adamc@214 1169 match n with
adamc@91 1170 | O => fun _ => Yes
adamc@91 1171 | S n'' => fun _ =>
adamc@91 1172 le_gt_dec (length s) n'
adam@338 1173 || dec_star'' (n := n') (star P)
adam@338 1174 (fun n0 _ => Reduce (F n'' n0 _)) (length s - n')
adamc@92 1175 end); clear F; crush; eauto;
adamc@92 1176 match goal with
adamc@92 1177 | [ H : star _ _ |- _ ] => apply star_substring_inv in H; crush; eauto
adamc@92 1178 end;
adamc@92 1179 match goal with
adamc@92 1180 | [ H1 : _ < _ - _, H2 : forall l' : nat, _ <= _ - _ -> _ |- _ ] =>
adamc@92 1181 generalize (H2 _ (lt_le_S _ _ H1)); tauto
adamc@92 1182 end.
adamc@91 1183 Defined.
adamc@91 1184
adam@380 1185 (** Finally, we have [dec_star], defined by straightforward reduction from [dec_star']. *)
adamc@93 1186
adamc@214 1187 Definition dec_star : {star P s} + {~ star P s}.
adam@380 1188 refine (Reduce (dec_star' (n := length s) 0 _)); crush.
adamc@91 1189 Defined.
adamc@91 1190 End dec_star.
adamc@91 1191
adamc@93 1192 (* begin hide *)
adamc@86 1193 Lemma app_cong : forall x1 y1 x2 y2,
adamc@86 1194 x1 = x2
adamc@86 1195 -> y1 = y2
adamc@86 1196 -> x1 ++ y1 = x2 ++ y2.
adamc@86 1197 congruence.
adamc@86 1198 Qed.
adamc@86 1199
adamc@86 1200 Hint Resolve app_cong.
adamc@93 1201 (* end hide *)
adamc@93 1202
adamc@93 1203 (** 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 1204
adam@297 1205 Definition matches : forall P (r : regexp P) s, {P s} + {~ P s}.
adamc@214 1206 refine (fix F P (r : regexp P) s : {P s} + {~ P s} :=
adamc@86 1207 match r with
adamc@86 1208 | Char ch => string_dec s (String ch "")
adamc@86 1209 | Concat _ _ r1 r2 => Reduce (split (F _ r1) (F _ r2) s)
adamc@87 1210 | Or _ _ r1 r2 => F _ r1 s || F _ r2 s
adamc@91 1211 | Star _ r => dec_star _ _ _
adamc@86 1212 end); crush;
adamc@86 1213 match goal with
adam@426 1214 | [ H : _ |- _ ] => generalize (H _ _ (eq_refl _))
adamc@93 1215 end; tauto.
adamc@86 1216 Defined.
adamc@86 1217
adam@283 1218 (** 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 1219
adamc@93 1220 (* begin hide *)
adamc@86 1221 Example hi := Concat (Char "h"%char) (Char "i"%char).
adam@380 1222 Eval hnf in matches hi "hi".
adam@380 1223 Eval hnf in matches hi "bye".
adamc@87 1224
adamc@87 1225 Example a_b := Or (Char "a"%char) (Char "b"%char).
adam@380 1226 Eval hnf in matches a_b "".
adam@380 1227 Eval hnf in matches a_b "a".
adam@380 1228 Eval hnf in matches a_b "aa".
adam@380 1229 Eval hnf in matches a_b "b".
adam@283 1230 (* end hide *)
adam@283 1231
adam@405 1232 (** 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. *)
adamc@91 1233
adamc@91 1234 Example a_star := Star (Char "a"%char).
adam@380 1235 Eval hnf in matches a_star "".
adam@380 1236 Eval hnf in matches a_star "a".
adam@380 1237 Eval hnf in matches a_star "b".
adam@380 1238 Eval hnf in matches a_star "aa".
adam@283 1239
adam@283 1240 (** 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. *)