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Fixes added while proofreading JFR camera-ready
author Adam Chlipala <adam@chlipala.net>
date Thu, 09 Dec 2010 13:44:57 -0500
parents 2c88fc1dbe33
children 559ec7328410
rev   line source
adam@283 1 (* Copyright (c) 2008-2010, 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
adamc@86 13 Require Import Tactics 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@292 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
adamc@83 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 which 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
adamc@84 28 (** Many introductions to dependent types start out by showing how to use them to eliminate 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, 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
adamc@84 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 sublist. We may apply [ilist] to any natural number, even natural numbers that are only known at runtime. It is this breaking of the %\textit{%#<i>#phase distinction#</i>#%}% that characterizes [ilist] as %\textit{%#<i>#dependently typed#</i>#%}%.
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
adamc@213 49 (** In Coq version 8.1 and earlier, this definition leads to an error message:
adamc@84 50
adamc@84 51 [[
adamc@84 52 The term "ls2" has type "ilist n2" while it is expected to have type
adamc@84 53 "ilist (?14 + n2)"
adamc@213 54
adamc@84 55 ]]
adamc@84 56
adamc@213 57 In Coq's core language, without explicit annotations, Coq does not enrich our typing assumptions in the branches of a [match] expression. It is clear that the unification variable [?14] should be resolved to 0 in this context, so that we have [0 + n2] reducing to [n2], but Coq does not realize that. We cannot fix the problem using just the simple [return] clauses we applied in the last chapter. We need to combine a [return] clause with a new kind of annotation, an [in] clause. This is exactly what the inference heuristics do in Coq 8.2 and later.
adamc@213 58
adamc@213 59 Specifically, Coq infers the following definition from the simpler one. *)
adamc@84 60
adamc@100 61 (* EX: Implement concatenation *)
adamc@100 62
adamc@100 63 (* begin thide *)
adamc@213 64 Fixpoint app' n1 (ls1 : ilist n1) n2 (ls2 : ilist n2) : ilist (n1 + n2) :=
adamc@84 65 match ls1 in (ilist n1) return (ilist (n1 + n2)) with
adamc@84 66 | Nil => ls2
adamc@213 67 | Cons _ x ls1' => Cons x (app' ls1' ls2)
adamc@84 68 end.
adamc@100 69 (* end thide *)
adamc@84 70
adamc@213 71 (** Using [return] alone allowed us to express a dependency of the [match] result type on the %\textit{%#<i>#value#</i>#%}% of the discriminee. What [in] adds to our arsenal is a way of expressing a dependency on the %\textit{%#<i>#type#</i>#%}% of the discriminee. Specifically, the [n1] in the [in] clause above is a %\textit{%#<i>#binding occurrence#</i>#%}% whose scope is the [return] clause.
adamc@84 72
adamc@84 73 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 %\textit{%#<i>#parameters#</i>#%}% 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 74
adam@283 75 Our [app] function could be typed in so-called %\textit{%#<i>#stratified#</i>#%}% 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 76
adamc@100 77 (* EX: Implement injection from normal lists *)
adamc@100 78
adamc@100 79 (* begin thide *)
adamc@84 80 Fixpoint inject (ls : list A) : ilist (length ls) :=
adamc@213 81 match ls with
adamc@84 82 | nil => Nil
adamc@84 83 | h :: t => Cons h (inject t)
adamc@84 84 end.
adamc@84 85
adamc@84 86 (** We can define an inverse conversion and prove that it really is an inverse. *)
adamc@84 87
adamc@213 88 Fixpoint unject n (ls : ilist n) : list A :=
adamc@84 89 match ls with
adamc@84 90 | Nil => nil
adamc@84 91 | Cons _ h t => h :: unject t
adamc@84 92 end.
adamc@84 93
adamc@84 94 Theorem inject_inverse : forall ls, unject (inject ls) = ls.
adamc@84 95 induction ls; crush.
adamc@84 96 Qed.
adamc@100 97 (* end thide *)
adamc@100 98
adamc@100 99 (* EX: Implement statically-checked "car"/"hd" *)
adamc@84 100
adam@283 101 (** 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 in the previous chapter.
adamc@84 102
adamc@84 103 [[
adamc@84 104 Definition hd n (ls : ilist (S n)) : A :=
adamc@84 105 match ls with
adamc@84 106 | Nil => ???
adamc@84 107 | Cons _ h _ => h
adamc@84 108 end.
adamc@213 109
adamc@213 110 ]]
adamc@84 111
adamc@84 112 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@84 113
adamc@84 114 [[
adamc@84 115 Definition hd n (ls : ilist (S n)) : A :=
adamc@84 116 match ls with
adamc@84 117 | Cons _ h _ => h
adamc@84 118 end.
adamc@84 119
adamc@84 120 Error: Non exhaustive pattern-matching: no clause found for pattern Nil
adamc@213 121
adamc@84 122 ]]
adamc@84 123
adam@275 124 Unlike in ML, we cannot use inexhaustive pattern matching, because there is no conception of a %\texttt{%#<tt>#Match#</tt>#%}% exception to be thrown. In fact, recent versions of Coq %\textit{%#<i>#do#</i>#%}% 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@84 125
adamc@84 126 [[
adamc@84 127 Definition hd n (ls : ilist (S n)) : A :=
adamc@84 128 match ls in (ilist (S n)) with
adamc@84 129 | Cons _ h _ => h
adamc@84 130 end.
adamc@84 131
adamc@84 132 Error: The reference n was not found in the current environment
adamc@213 133
adamc@84 134 ]]
adamc@84 135
adamc@84 136 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, which is undecidable. There %\textit{%#<i>#are#</i>#%}% 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@84 137
adamc@84 138 Our final, working attempt at [hd] uses an auxiliary function and a surprising [return] annotation. *)
adamc@84 139
adamc@100 140 (* begin thide *)
adamc@84 141 Definition hd' n (ls : ilist n) :=
adamc@84 142 match ls in (ilist n) return (match n with O => unit | S _ => A end) with
adamc@84 143 | Nil => tt
adamc@84 144 | Cons _ h _ => h
adamc@84 145 end.
adamc@84 146
adam@283 147 Check hd'.
adam@283 148 (** %\vspace{-.15in}% [[
adam@283 149 hd'
adam@283 150 : forall n : nat, ilist n -> match n with
adam@283 151 | 0 => unit
adam@283 152 | S _ => A
adam@283 153 end
adam@283 154
adam@283 155 ]] *)
adam@283 156
adamc@84 157 Definition hd n (ls : ilist (S n)) : A := hd' ls.
adamc@100 158 (* end thide *)
adamc@84 159
adamc@84 160 (** 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@84 161
adamc@84 162 End ilist.
adamc@85 163
adamc@85 164
adamc@85 165 (** * A Tagless Interpreter *)
adamc@85 166
adamc@85 167 (** 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 %\textit{%#<i>#tagless#</i>#%}% interpreter that both removes this source of runtime ineffiency and gives us more confidence that our implementation is correct. *)
adamc@85 168
adamc@85 169 Inductive type : Set :=
adamc@85 170 | Nat : type
adamc@85 171 | Bool : type
adamc@85 172 | Prod : type -> type -> type.
adamc@85 173
adamc@85 174 Inductive exp : type -> Set :=
adamc@85 175 | NConst : nat -> exp Nat
adamc@85 176 | Plus : exp Nat -> exp Nat -> exp Nat
adamc@85 177 | Eq : exp Nat -> exp Nat -> exp Bool
adamc@85 178
adamc@85 179 | BConst : bool -> exp Bool
adamc@85 180 | And : exp Bool -> exp Bool -> exp Bool
adamc@85 181 | If : forall t, exp Bool -> exp t -> exp t -> exp t
adamc@85 182
adamc@85 183 | Pair : forall t1 t2, exp t1 -> exp t2 -> exp (Prod t1 t2)
adamc@85 184 | Fst : forall t1 t2, exp (Prod t1 t2) -> exp t1
adamc@85 185 | Snd : forall t1 t2, exp (Prod t1 t2) -> exp t2.
adamc@85 186
adamc@85 187 (** 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@85 188
adamc@85 189 We can give types and expressions semantics in a new style, based critically on the chance for %\textit{%#<i>#type-level computation#</i>#%}%. *)
adamc@85 190
adamc@85 191 Fixpoint typeDenote (t : type) : Set :=
adamc@85 192 match t with
adamc@85 193 | Nat => nat
adamc@85 194 | Bool => bool
adamc@85 195 | Prod t1 t2 => typeDenote t1 * typeDenote t2
adamc@85 196 end%type.
adamc@85 197
adam@292 198 (** [typeDenote] compiles types of our object language into %``%#"#native#"#%''% Coq types. It is deceptively easy to implement. The only new thing we see is the [%type] 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. [type] is one example of an identifer bound to a %\textit{%#<i>#notation scope#</i>#%}%. We will deal more explicitly with notations and notation scopes in later chapters.
adamc@85 199
adamc@85 200 We can define a function [expDenote] that is typed in terms of [typeDenote]. *)
adamc@85 201
adamc@213 202 Fixpoint expDenote t (e : exp t) : typeDenote t :=
adamc@213 203 match e with
adamc@85 204 | NConst n => n
adamc@85 205 | Plus e1 e2 => expDenote e1 + expDenote e2
adamc@85 206 | Eq e1 e2 => if eq_nat_dec (expDenote e1) (expDenote e2) then true else false
adamc@85 207
adamc@85 208 | BConst b => b
adamc@85 209 | And e1 e2 => expDenote e1 && expDenote e2
adamc@85 210 | If _ e' e1 e2 => if expDenote e' then expDenote e1 else expDenote e2
adamc@85 211
adamc@85 212 | Pair _ _ e1 e2 => (expDenote e1, expDenote e2)
adamc@85 213 | Fst _ _ e' => fst (expDenote e')
adamc@85 214 | Snd _ _ e' => snd (expDenote e')
adamc@85 215 end.
adamc@85 216
adamc@213 217 (** 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@85 218
adamc@85 219 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@85 220
adamc@85 221 [[
adamc@85 222 Definition pairOut t1 t2 (e : exp (Prod t1 t2)) : option (exp t1 * exp t2) :=
adamc@85 223 match e in (exp (Prod t1 t2)) return option (exp t1 * exp t2) with
adamc@85 224 | Pair _ _ e1 e2 => Some (e1, e2)
adamc@85 225 | _ => None
adamc@85 226 end.
adamc@85 227
adamc@85 228 Error: The reference t2 was not found in the current environment
adamc@213 229 ]]
adamc@85 230
adamc@85 231 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@85 232
adamc@100 233 (* EX: Define a function [pairOut : forall t1 t2, exp (Prod t1 t2) -> option (exp t1 * exp t2)] *)
adamc@100 234
adamc@100 235 (* begin thide *)
adamc@85 236 Definition pairOutType (t : type) :=
adamc@85 237 match t with
adamc@85 238 | Prod t1 t2 => option (exp t1 * exp t2)
adamc@85 239 | _ => unit
adamc@85 240 end.
adamc@85 241
adamc@85 242 (** 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@85 243
adamc@85 244 Definition pairOutDefault (t : type) :=
adamc@85 245 match t return (pairOutType t) with
adamc@85 246 | Prod _ _ => None
adamc@85 247 | _ => tt
adamc@85 248 end.
adamc@85 249
adamc@85 250 (** Now [pairOut] is deceptively easy to write. *)
adamc@85 251
adamc@85 252 Definition pairOut t (e : exp t) :=
adamc@85 253 match e in (exp t) return (pairOutType t) with
adamc@85 254 | Pair _ _ e1 e2 => Some (e1, e2)
adamc@85 255 | _ => pairOutDefault _
adamc@85 256 end.
adamc@100 257 (* end thide *)
adamc@85 258
adamc@85 259 (** 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, but is similar to what goes on with Haskell type classes.
adamc@85 260
adamc@213 261 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 by writing [return _]. *)
adamc@85 262
adamc@204 263 Fixpoint cfold t (e : exp t) : exp t :=
adamc@204 264 match e with
adamc@85 265 | NConst n => NConst n
adamc@85 266 | Plus e1 e2 =>
adamc@85 267 let e1' := cfold e1 in
adamc@85 268 let e2' := cfold e2 in
adamc@204 269 match e1', e2' return _ with
adamc@85 270 | NConst n1, NConst n2 => NConst (n1 + n2)
adamc@85 271 | _, _ => Plus e1' e2'
adamc@85 272 end
adamc@85 273 | Eq e1 e2 =>
adamc@85 274 let e1' := cfold e1 in
adamc@85 275 let e2' := cfold e2 in
adamc@204 276 match e1', e2' return _ with
adamc@85 277 | NConst n1, NConst n2 => BConst (if eq_nat_dec n1 n2 then true else false)
adamc@85 278 | _, _ => Eq e1' e2'
adamc@85 279 end
adamc@85 280
adamc@85 281 | BConst b => BConst b
adamc@85 282 | And e1 e2 =>
adamc@85 283 let e1' := cfold e1 in
adamc@85 284 let e2' := cfold e2 in
adamc@204 285 match e1', e2' return _ with
adamc@85 286 | BConst b1, BConst b2 => BConst (b1 && b2)
adamc@85 287 | _, _ => And e1' e2'
adamc@85 288 end
adamc@85 289 | If _ e e1 e2 =>
adamc@85 290 let e' := cfold e in
adamc@85 291 match e' with
adamc@85 292 | BConst true => cfold e1
adamc@85 293 | BConst false => cfold e2
adamc@85 294 | _ => If e' (cfold e1) (cfold e2)
adamc@85 295 end
adamc@85 296
adamc@85 297 | Pair _ _ e1 e2 => Pair (cfold e1) (cfold e2)
adamc@85 298 | Fst _ _ e =>
adamc@85 299 let e' := cfold e in
adamc@85 300 match pairOut e' with
adamc@85 301 | Some p => fst p
adamc@85 302 | None => Fst e'
adamc@85 303 end
adamc@85 304 | Snd _ _ e =>
adamc@85 305 let e' := cfold e in
adamc@85 306 match pairOut e' with
adamc@85 307 | Some p => snd p
adamc@85 308 | None => Snd e'
adamc@85 309 end
adamc@85 310 end.
adamc@85 311
adamc@85 312 (** The correctness theorem for [cfold] turns out to be easy to prove, once we get over one serious hurdle. *)
adamc@85 313
adamc@85 314 Theorem cfold_correct : forall t (e : exp t), expDenote e = expDenote (cfold e).
adamc@100 315 (* begin thide *)
adamc@85 316 induction e; crush.
adamc@85 317
adamc@85 318 (** The first remaining subgoal is:
adamc@85 319
adamc@85 320 [[
adamc@85 321 expDenote (cfold e1) + expDenote (cfold e2) =
adamc@85 322 expDenote
adamc@85 323 match cfold e1 with
adamc@85 324 | NConst n1 =>
adamc@85 325 match cfold e2 with
adamc@85 326 | NConst n2 => NConst (n1 + n2)
adamc@85 327 | Plus _ _ => Plus (cfold e1) (cfold e2)
adamc@85 328 | Eq _ _ => Plus (cfold e1) (cfold e2)
adamc@85 329 | BConst _ => Plus (cfold e1) (cfold e2)
adamc@85 330 | And _ _ => Plus (cfold e1) (cfold e2)
adamc@85 331 | If _ _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 332 | Pair _ _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 333 | Fst _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 334 | Snd _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 335 end
adamc@85 336 | Plus _ _ => Plus (cfold e1) (cfold e2)
adamc@85 337 | Eq _ _ => Plus (cfold e1) (cfold e2)
adamc@85 338 | BConst _ => Plus (cfold e1) (cfold e2)
adamc@85 339 | And _ _ => Plus (cfold e1) (cfold e2)
adamc@85 340 | If _ _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 341 | Pair _ _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 342 | Fst _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 343 | Snd _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 344 end
adamc@213 345
adamc@85 346 ]]
adamc@85 347
adamc@85 348 We would like to do a case analysis on [cfold e1], and we attempt that in the way that has worked so far.
adamc@85 349
adamc@85 350 [[
adamc@85 351 destruct (cfold e1).
adamc@85 352
adamc@85 353 User error: e1 is used in hypothesis e
adamc@213 354
adamc@85 355 ]]
adamc@85 356
adamc@85 357 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@85 358
adamc@213 359 For our current proof, we can use a tactic [dep_destruct] defined in the book [Tactics] 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. [dep_destruct] makes a best effort to handle some common cases, relying upon the more primitive [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. *)
adamc@85 360
adamc@85 361 dep_destruct (cfold e1).
adamc@85 362
adamc@85 363 (** 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@85 364
adamc@213 365 This is the only new trick we need to learn to complete the proof. We can back up and give a short, automated proof. The main inconvenience in the proof is that we cannot write a pattern that matches a [match] without including a case for every constructor of the inductive type we match over. *)
adamc@85 366
adamc@85 367 Restart.
adamc@85 368
adamc@85 369 induction e; crush;
adamc@85 370 repeat (match goal with
adamc@213 371 | [ |- context[match cfold ?E with NConst _ => _ | Plus _ _ => _
adamc@213 372 | Eq _ _ => _ | BConst _ => _ | And _ _ => _
adamc@213 373 | If _ _ _ _ => _ | Pair _ _ _ _ => _
adamc@213 374 | Fst _ _ _ => _ | Snd _ _ _ => _ end] ] =>
adamc@213 375 dep_destruct (cfold E)
adamc@213 376 | [ |- context[match pairOut (cfold ?E) with Some _ => _
adamc@213 377 | None => _ end] ] =>
adamc@213 378 dep_destruct (cfold E)
adamc@85 379 | [ |- (if ?E then _ else _) = _ ] => destruct E
adamc@85 380 end; crush).
adamc@85 381 Qed.
adamc@100 382 (* end thide *)
adamc@86 383
adamc@86 384
adamc@103 385 (** * Dependently-Typed Red-Black Trees *)
adamc@94 386
adamc@214 387 (** 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@100 388
adamc@94 389 Inductive color : Set := Red | Black.
adamc@94 390
adamc@94 391 Inductive rbtree : color -> nat -> Set :=
adamc@94 392 | Leaf : rbtree Black 0
adamc@214 393 | RedNode : forall n, rbtree Black n -> nat -> rbtree Black n -> rbtree Red n
adamc@94 394 | BlackNode : forall c1 c2 n, rbtree c1 n -> nat -> rbtree c2 n -> rbtree Black (S n).
adamc@94 395
adamc@214 396 (** 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@214 397
adamc@214 398 (** 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@214 399
adamc@100 400 (* EX: Prove that every [rbtree] is balanced. *)
adamc@100 401
adamc@100 402 (* begin thide *)
adamc@95 403 Require Import Max Min.
adamc@95 404
adamc@95 405 Section depth.
adamc@95 406 Variable f : nat -> nat -> nat.
adamc@95 407
adamc@214 408 Fixpoint depth c n (t : rbtree c n) : nat :=
adamc@95 409 match t with
adamc@95 410 | Leaf => 0
adamc@95 411 | RedNode _ t1 _ t2 => S (f (depth t1) (depth t2))
adamc@95 412 | BlackNode _ _ _ t1 _ t2 => S (f (depth t1) (depth t2))
adamc@95 413 end.
adamc@95 414 End depth.
adamc@95 415
adamc@214 416 (** 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@214 417
adam@283 418 Check min_dec.
adam@283 419 (** %\vspace{-.15in}% [[
adam@283 420 min_dec
adam@283 421 : forall n m : nat, {min n m = n} + {min n m = m}
adam@283 422
adam@283 423 ]] *)
adam@283 424
adamc@95 425 Theorem depth_min : forall c n (t : rbtree c n), depth min t >= n.
adamc@95 426 induction t; crush;
adamc@95 427 match goal with
adamc@95 428 | [ |- context[min ?X ?Y] ] => destruct (min_dec X Y)
adamc@95 429 end; crush.
adamc@95 430 Qed.
adamc@95 431
adamc@214 432 (** There is an analogous upper-bound theorem based on black depth. Unfortunately, a symmetric proof script does not suffice to establish it. *)
adamc@214 433
adamc@214 434 Theorem depth_max : forall c n (t : rbtree c n), depth max t <= 2 * n + 1.
adamc@214 435 induction t; crush;
adamc@214 436 match goal with
adamc@214 437 | [ |- context[max ?X ?Y] ] => destruct (max_dec X Y)
adamc@214 438 end; crush.
adamc@214 439
adamc@214 440 (** Two subgoals remain. One of them is: [[
adamc@214 441 n : nat
adamc@214 442 t1 : rbtree Black n
adamc@214 443 n0 : nat
adamc@214 444 t2 : rbtree Black n
adamc@214 445 IHt1 : depth max t1 <= n + (n + 0) + 1
adamc@214 446 IHt2 : depth max t2 <= n + (n + 0) + 1
adamc@214 447 e : max (depth max t1) (depth max t2) = depth max t1
adamc@214 448 ============================
adamc@214 449 S (depth max t1) <= n + (n + 0) + 1
adamc@214 450
adamc@214 451 ]]
adamc@214 452
adamc@214 453 We see that [IHt1] is %\textit{%#<i>#almost#</i>#%}% 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@214 454
adamc@214 455 Abort.
adamc@214 456
adamc@214 457 (** 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@214 458
adamc@95 459 Lemma depth_max' : forall c n (t : rbtree c n), match c with
adamc@95 460 | Red => depth max t <= 2 * n + 1
adamc@95 461 | Black => depth max t <= 2 * n
adamc@95 462 end.
adamc@95 463 induction t; crush;
adamc@95 464 match goal with
adamc@95 465 | [ |- context[max ?X ?Y] ] => destruct (max_dec X Y)
adamc@100 466 end; crush;
adamc@100 467 repeat (match goal with
adamc@214 468 | [ H : context[match ?C with Red => _ | Black => _ end] |- _ ] =>
adamc@214 469 destruct C
adamc@100 470 end; crush).
adamc@95 471 Qed.
adamc@95 472
adamc@214 473 (** The original theorem follows easily from the lemma. We use the tactic [generalize pf], which, when [pf] proves the proposition [P], changes the goal from [Q] to [P -> Q]. It is useful to do this 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@214 474
adamc@95 475 Theorem depth_max : forall c n (t : rbtree c n), depth max t <= 2 * n + 1.
adamc@95 476 intros; generalize (depth_max' t); destruct c; crush.
adamc@95 477 Qed.
adamc@95 478
adamc@214 479 (** 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@214 480
adamc@95 481 Theorem balanced : forall c n (t : rbtree c n), 2 * depth min t + 1 >= depth max t.
adamc@95 482 intros; generalize (depth_min t); generalize (depth_max t); crush.
adamc@95 483 Qed.
adamc@100 484 (* end thide *)
adamc@95 485
adamc@214 486 (** 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@95 487
adamc@94 488 Inductive rtree : nat -> Set :=
adamc@94 489 | RedNode' : forall c1 c2 n, rbtree c1 n -> nat -> rbtree c2 n -> rtree n.
adamc@94 490
adamc@214 491 (** 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@214 492
adamc@96 493 Section present.
adamc@96 494 Variable x : nat.
adamc@96 495
adamc@214 496 Fixpoint present c n (t : rbtree c n) : Prop :=
adamc@96 497 match t with
adamc@96 498 | Leaf => False
adamc@96 499 | RedNode _ a y b => present a \/ x = y \/ present b
adamc@96 500 | BlackNode _ _ _ a y b => present a \/ x = y \/ present b
adamc@96 501 end.
adamc@96 502
adamc@96 503 Definition rpresent n (t : rtree n) : Prop :=
adamc@96 504 match t with
adamc@96 505 | RedNode' _ _ _ a y b => present a \/ x = y \/ present b
adamc@96 506 end.
adamc@96 507 End present.
adamc@96 508
adamc@214 509 (** 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 [sigT] type fills this role. *)
adamc@214 510
adamc@100 511 Locate "{ _ : _ & _ }".
adamc@214 512 (** [[
adamc@214 513 Notation Scope
adamc@214 514 "{ x : A & P }" := sigT (fun x : A => P)
adamc@214 515 ]] *)
adamc@214 516
adamc@100 517 Print sigT.
adamc@214 518 (** [[
adamc@214 519 Inductive sigT (A : Type) (P : A -> Type) : Type :=
adamc@214 520 existT : forall x : A, P x -> sigT P
adamc@214 521 ]] *)
adamc@214 522
adamc@214 523 (** It will be helpful to define a concise notation for the constructor of [sigT]. *)
adamc@100 524
adamc@94 525 Notation "{< x >}" := (existT _ _ x).
adamc@94 526
adamc@214 527 (** 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@214 528
adamc@214 529 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]. *)
adamc@214 530
adamc@94 531 Definition balance1 n (a : rtree n) (data : nat) c2 :=
adamc@214 532 match a in rtree n return rbtree c2 n
adamc@214 533 -> { c : color & rbtree c (S n) } with
adamc@94 534 | RedNode' _ _ _ t1 y t2 =>
adamc@214 535 match t1 in rbtree c n return rbtree _ n -> rbtree c2 n
adamc@214 536 -> { c : color & rbtree c (S n) } with
adamc@214 537 | RedNode _ a x b => fun c d =>
adamc@214 538 {<RedNode (BlackNode a x b) y (BlackNode c data d)>}
adamc@94 539 | t1' => fun t2 =>
adamc@214 540 match t2 in rbtree c n return rbtree _ n -> rbtree c2 n
adamc@214 541 -> { c : color & rbtree c (S n) } with
adamc@214 542 | RedNode _ b x c => fun a d =>
adamc@214 543 {<RedNode (BlackNode a y b) x (BlackNode c data d)>}
adamc@95 544 | b => fun a t => {<BlackNode (RedNode a y b) data t>}
adamc@94 545 end t1'
adamc@94 546 end t2
adamc@94 547 end.
adamc@94 548
adamc@214 549 (** We apply a trick that I call the %\textit{%#<i>#convoy pattern#</i>#%}%. Recall that [match] annotations only make it possible to describe a dependence of a [match] %\textit{%#<i>#result type#</i>#%}% 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@214 550
adam@292 551 In particular, we can extend the [match] to return %\textit{%#<i>#functions over the free variables whose types we want to refine#</i>#%}%. 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@214 552
adam@292 553 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#"#%''% 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@214 554
adamc@214 555 Here is the symmetric function [balance2], for cases where the possibly-invalid tree appears on the right rather than on the left. *)
adamc@214 556
adamc@94 557 Definition balance2 n (a : rtree n) (data : nat) c2 :=
adamc@94 558 match a in rtree n return rbtree c2 n -> { c : color & rbtree c (S n) } with
adamc@94 559 | RedNode' _ _ _ t1 z t2 =>
adamc@214 560 match t1 in rbtree c n return rbtree _ n -> rbtree c2 n
adamc@214 561 -> { c : color & rbtree c (S n) } with
adamc@214 562 | RedNode _ b y c => fun d a =>
adamc@214 563 {<RedNode (BlackNode a data b) y (BlackNode c z d)>}
adamc@94 564 | t1' => fun t2 =>
adamc@214 565 match t2 in rbtree c n return rbtree _ n -> rbtree c2 n
adamc@214 566 -> { c : color & rbtree c (S n) } with
adamc@214 567 | RedNode _ c z' d => fun b a =>
adamc@214 568 {<RedNode (BlackNode a data b) z (BlackNode c z' d)>}
adamc@95 569 | b => fun a t => {<BlackNode t data (RedNode a z b)>}
adamc@94 570 end t1'
adamc@94 571 end t2
adamc@94 572 end.
adamc@94 573
adamc@214 574 (** 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@214 575
adamc@94 576 Section insert.
adamc@94 577 Variable x : nat.
adamc@94 578
adamc@214 579 (** 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@214 580
adamc@94 581 Definition insResult c n :=
adamc@94 582 match c with
adamc@94 583 | Red => rtree n
adamc@94 584 | Black => { c' : color & rbtree c' n }
adamc@94 585 end.
adamc@94 586
adamc@214 587 (** 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 arbitary color.
adamc@214 588
adamc@214 589 Here is the definition of [ins]. Again, we do not want to dwell on the functional details. *)
adamc@214 590
adamc@214 591 Fixpoint ins c n (t : rbtree c n) : insResult c n :=
adamc@214 592 match t with
adamc@94 593 | Leaf => {< RedNode Leaf x Leaf >}
adamc@94 594 | RedNode _ a y b =>
adamc@94 595 if le_lt_dec x y
adamc@94 596 then RedNode' (projT2 (ins a)) y b
adamc@94 597 else RedNode' a y (projT2 (ins b))
adamc@94 598 | BlackNode c1 c2 _ a y b =>
adamc@94 599 if le_lt_dec x y
adamc@94 600 then
adamc@94 601 match c1 return insResult c1 _ -> _ with
adamc@94 602 | Red => fun ins_a => balance1 ins_a y b
adamc@94 603 | _ => fun ins_a => {< BlackNode (projT2 ins_a) y b >}
adamc@94 604 end (ins a)
adamc@94 605 else
adamc@94 606 match c2 return insResult c2 _ -> _ with
adamc@94 607 | Red => fun ins_b => balance2 ins_b y a
adamc@94 608 | _ => fun ins_b => {< BlackNode a y (projT2 ins_b) >}
adamc@94 609 end (ins b)
adamc@94 610 end.
adamc@94 611
adamc@214 612 (** 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 satisifies 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 %\textit{%#<i>#the result of a recursive call#</i>#%}%, rather than just on that call's argument.
adamc@214 613
adamc@214 614 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@214 615
adamc@94 616 Definition insertResult c n :=
adamc@94 617 match c with
adamc@94 618 | Red => rbtree Black (S n)
adamc@94 619 | Black => { c' : color & rbtree c' n }
adamc@94 620 end.
adamc@94 621
adamc@214 622 (** A simple clean-up procedure translates [insResult]s into [insertResult]s. *)
adamc@214 623
adamc@97 624 Definition makeRbtree c n : insResult c n -> insertResult c n :=
adamc@214 625 match c with
adamc@94 626 | Red => fun r =>
adamc@214 627 match r with
adamc@94 628 | RedNode' _ _ _ a x b => BlackNode a x b
adamc@94 629 end
adamc@94 630 | Black => fun r => r
adamc@94 631 end.
adamc@94 632
adamc@214 633 (** 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@214 634
adamc@97 635 Implicit Arguments makeRbtree [c n].
adamc@94 636
adamc@214 637 (** Finally, we define [insert] as a simple composition of [ins] and [makeRbtree]. *)
adamc@214 638
adamc@94 639 Definition insert c n (t : rbtree c n) : insertResult c n :=
adamc@97 640 makeRbtree (ins t).
adamc@94 641
adamc@214 642 (** 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@214 643
adamc@95 644 Section present.
adamc@95 645 Variable z : nat.
adamc@95 646
adamc@214 647 (** 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@214 648
adamc@214 649 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 [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@214 650
adamc@98 651 Ltac present_balance :=
adamc@98 652 crush;
adamc@98 653 repeat (match goal with
adamc@98 654 | [ H : context[match ?T with
adamc@98 655 | Leaf => _
adamc@98 656 | RedNode _ _ _ _ => _
adamc@98 657 | BlackNode _ _ _ _ _ _ => _
adamc@98 658 end] |- _ ] => dep_destruct T
adamc@98 659 | [ |- context[match ?T with
adamc@98 660 | Leaf => _
adamc@98 661 | RedNode _ _ _ _ => _
adamc@98 662 | BlackNode _ _ _ _ _ _ => _
adamc@98 663 end] ] => dep_destruct T
adamc@98 664 end; crush).
adamc@98 665
adamc@214 666 (** 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@214 667
adam@294 668 Lemma present_balance1 : forall n (a : rtree n) (y : nat) c2 (b : rbtree c2 n),
adamc@95 669 present z (projT2 (balance1 a y b))
adamc@95 670 <-> rpresent z a \/ z = y \/ present z b.
adamc@98 671 destruct a; present_balance.
adamc@95 672 Qed.
adamc@95 673
adamc@213 674 Lemma present_balance2 : forall n (a : rtree n) (y : nat) c2 (b : rbtree c2 n),
adamc@95 675 present z (projT2 (balance2 a y b))
adamc@95 676 <-> rpresent z a \/ z = y \/ present z b.
adamc@98 677 destruct a; present_balance.
adamc@95 678 Qed.
adamc@95 679
adamc@214 680 (** 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@214 681
adamc@95 682 Definition present_insResult c n :=
adamc@95 683 match c return (rbtree c n -> insResult c n -> Prop) with
adamc@95 684 | Red => fun t r => rpresent z r <-> z = x \/ present z t
adamc@95 685 | Black => fun t r => present z (projT2 r) <-> z = x \/ present z t
adamc@95 686 end.
adamc@95 687
adamc@214 688 (** 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@214 689
adamc@214 690 (** printing * $*$ *)
adamc@214 691
adamc@95 692 Theorem present_ins : forall c n (t : rbtree c n),
adamc@95 693 present_insResult t (ins t).
adamc@95 694 induction t; crush;
adamc@95 695 repeat (match goal with
adamc@95 696 | [ H : context[if ?E then _ else _] |- _ ] => destruct E
adamc@95 697 | [ |- context[if ?E then _ else _] ] => destruct E
adamc@214 698 | [ H : context[match ?C with Red => _ | Black => _ end]
adamc@214 699 |- _ ] => destruct C
adamc@95 700 end; crush);
adamc@95 701 try match goal with
adamc@95 702 | [ H : context[balance1 ?A ?B ?C] |- _ ] =>
adamc@95 703 generalize (present_balance1 A B C)
adamc@95 704 end;
adamc@95 705 try match goal with
adamc@95 706 | [ H : context[balance2 ?A ?B ?C] |- _ ] =>
adamc@95 707 generalize (present_balance2 A B C)
adamc@95 708 end;
adamc@95 709 try match goal with
adamc@95 710 | [ |- context[balance1 ?A ?B ?C] ] =>
adamc@95 711 generalize (present_balance1 A B C)
adamc@95 712 end;
adamc@95 713 try match goal with
adamc@95 714 | [ |- context[balance2 ?A ?B ?C] ] =>
adamc@95 715 generalize (present_balance2 A B C)
adamc@95 716 end;
adamc@214 717 crush;
adamc@95 718 match goal with
adamc@95 719 | [ z : nat, x : nat |- _ ] =>
adamc@95 720 match goal with
adamc@95 721 | [ H : z = x |- _ ] => rewrite H in *; clear H
adamc@95 722 end
adamc@95 723 end;
adamc@95 724 tauto.
adamc@95 725 Qed.
adamc@95 726
adamc@214 727 (** printing * $\times$ *)
adamc@214 728
adamc@214 729 (** 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@214 730
adamc@213 731 Ltac present_insert :=
adamc@213 732 unfold insert; intros n t; inversion t;
adamc@97 733 generalize (present_ins t); simpl;
adamc@97 734 dep_destruct (ins t); tauto.
adamc@97 735
adamc@95 736 Theorem present_insert_Red : forall n (t : rbtree Red n),
adamc@95 737 present z (insert t)
adamc@95 738 <-> (z = x \/ present z t).
adamc@213 739 present_insert.
adamc@95 740 Qed.
adamc@95 741
adamc@95 742 Theorem present_insert_Black : forall n (t : rbtree Black n),
adamc@95 743 present z (projT2 (insert t))
adamc@95 744 <-> (z = x \/ present z t).
adamc@213 745 present_insert.
adamc@95 746 Qed.
adamc@95 747 End present.
adamc@94 748 End insert.
adamc@94 749
adam@283 750 (** We can generate executable OCaml code with the command [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 %\texttt{%#<tt>#Obj.magic#</tt>#%}%, 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 %\textit{%#<i>#value#</i>#%}% 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@283 751
adamc@94 752
adamc@86 753 (** * A Certified Regular Expression Matcher *)
adamc@86 754
adamc@93 755 (** 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@93 756
adam@283 757 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 [Strings] 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. *)
adamc@93 758
adamc@86 759 Require Import Ascii String.
adamc@86 760 Open Scope string_scope.
adamc@86 761
adamc@91 762 Section star.
adamc@91 763 Variable P : string -> Prop.
adamc@91 764
adamc@91 765 Inductive star : string -> Prop :=
adamc@91 766 | Empty : star ""
adamc@91 767 | Iter : forall s1 s2,
adamc@91 768 P s1
adamc@91 769 -> star s2
adamc@91 770 -> star (s1 ++ s2).
adamc@91 771 End star.
adamc@91 772
adam@283 773 (** 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@93 774
adamc@93 775 [[
adamc@93 776 Inductive regexp : (string -> Prop) -> Set :=
adamc@93 777 | Char : forall ch : ascii,
adamc@93 778 regexp (fun s => s = String ch "")
adamc@93 779 | Concat : forall (P1 P2 : string -> Prop) (r1 : regexp P1) (r2 : regexp P2),
adamc@93 780 regexp (fun s => exists s1, exists s2, s = s1 ++ s2 /\ P1 s1 /\ P2 s2).
adamc@93 781
adamc@93 782 User error: Large non-propositional inductive types must be in Type
adamc@214 783
adamc@93 784 ]]
adamc@93 785
adamc@93 786 What is a 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@93 787
adamc@93 788 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@93 789
adamc@89 790 Inductive regexp : (string -> Prop) -> Type :=
adamc@86 791 | Char : forall ch : ascii,
adamc@86 792 regexp (fun s => s = String ch "")
adamc@86 793 | Concat : forall P1 P2 (r1 : regexp P1) (r2 : regexp P2),
adamc@87 794 regexp (fun s => exists s1, exists s2, s = s1 ++ s2 /\ P1 s1 /\ P2 s2)
adamc@87 795 | Or : forall P1 P2 (r1 : regexp P1) (r2 : regexp P2),
adamc@91 796 regexp (fun s => P1 s \/ P2 s)
adamc@91 797 | Star : forall P (r : regexp P),
adamc@91 798 regexp (star P).
adamc@86 799
adamc@93 800 (** Many theorems about strings are useful for implementing a certified regexp matcher, and few of them are in the [Strings] library. The book source includes statements, proofs, and hint commands for a handful of such omittted theorems. Since they are orthogonal to our use of dependent types, we hide them in the rendered versions of this book. *)
adamc@93 801
adamc@93 802 (* begin hide *)
adamc@86 803 Open Scope specif_scope.
adamc@86 804
adamc@86 805 Lemma length_emp : length "" <= 0.
adamc@86 806 crush.
adamc@86 807 Qed.
adamc@86 808
adamc@86 809 Lemma append_emp : forall s, s = "" ++ s.
adamc@86 810 crush.
adamc@86 811 Qed.
adamc@86 812
adamc@86 813 Ltac substring :=
adamc@86 814 crush;
adamc@86 815 repeat match goal with
adamc@86 816 | [ |- context[match ?N with O => _ | S _ => _ end] ] => destruct N; crush
adamc@86 817 end.
adamc@86 818
adamc@86 819 Lemma substring_le : forall s n m,
adamc@86 820 length (substring n m s) <= m.
adamc@86 821 induction s; substring.
adamc@86 822 Qed.
adamc@86 823
adamc@86 824 Lemma substring_all : forall s,
adamc@86 825 substring 0 (length s) s = s.
adamc@86 826 induction s; substring.
adamc@86 827 Qed.
adamc@86 828
adamc@86 829 Lemma substring_none : forall s n,
adamc@93 830 substring n 0 s = "".
adamc@86 831 induction s; substring.
adamc@86 832 Qed.
adamc@86 833
adamc@86 834 Hint Rewrite substring_all substring_none : cpdt.
adamc@86 835
adamc@86 836 Lemma substring_split : forall s m,
adamc@86 837 substring 0 m s ++ substring m (length s - m) s = s.
adamc@86 838 induction s; substring.
adamc@86 839 Qed.
adamc@86 840
adamc@86 841 Lemma length_app1 : forall s1 s2,
adamc@86 842 length s1 <= length (s1 ++ s2).
adamc@86 843 induction s1; crush.
adamc@86 844 Qed.
adamc@86 845
adamc@86 846 Hint Resolve length_emp append_emp substring_le substring_split length_app1.
adamc@86 847
adamc@86 848 Lemma substring_app_fst : forall s2 s1 n,
adamc@86 849 length s1 = n
adamc@86 850 -> substring 0 n (s1 ++ s2) = s1.
adamc@86 851 induction s1; crush.
adamc@86 852 Qed.
adamc@86 853
adamc@86 854 Lemma substring_app_snd : forall s2 s1 n,
adamc@86 855 length s1 = n
adamc@86 856 -> substring n (length (s1 ++ s2) - n) (s1 ++ s2) = s2.
adamc@86 857 Hint Rewrite <- minus_n_O : cpdt.
adamc@86 858
adamc@86 859 induction s1; crush.
adamc@86 860 Qed.
adamc@86 861
adamc@214 862 Hint Rewrite substring_app_fst substring_app_snd using solve [trivial] : cpdt.
adamc@93 863 (* end hide *)
adamc@93 864
adamc@93 865 (** A few auxiliary functions help us in our final matcher definition. The function [split] will be used to implement the regexp concatenation case. *)
adamc@86 866
adamc@86 867 Section split.
adamc@86 868 Variables P1 P2 : string -> Prop.
adamc@214 869 Variable P1_dec : forall s, {P1 s} + {~ P1 s}.
adamc@214 870 Variable P2_dec : forall s, {P2 s} + {~ P2 s}.
adamc@93 871 (** We require a choice of two arbitrary string predicates and functions for deciding them. *)
adamc@86 872
adamc@86 873 Variable s : string.
adamc@93 874 (** 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@93 875
adamc@93 876 (** [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. [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@86 877
adamc@86 878 Definition split' (n : nat) : n <= length s
adamc@86 879 -> {exists s1, exists s2, length s1 <= n /\ s1 ++ s2 = s /\ P1 s1 /\ P2 s2}
adamc@214 880 + {forall s1 s2, length s1 <= n -> s1 ++ s2 = s -> ~ P1 s1 \/ ~ P2 s2}.
adamc@86 881 refine (fix F (n : nat) : n <= length s
adamc@86 882 -> {exists s1, exists s2, length s1 <= n /\ s1 ++ s2 = s /\ P1 s1 /\ P2 s2}
adamc@214 883 + {forall s1 s2, length s1 <= n -> s1 ++ s2 = s -> ~ P1 s1 \/ ~ P2 s2} :=
adamc@214 884 match n with
adamc@86 885 | O => fun _ => Reduce (P1_dec "" && P2_dec s)
adamc@93 886 | S n' => fun _ => (P1_dec (substring 0 (S n') s)
adamc@93 887 && P2_dec (substring (S n') (length s - S n') s))
adamc@86 888 || F n' _
adamc@86 889 end); clear F; crush; eauto 7;
adamc@86 890 match goal with
adamc@86 891 | [ _ : length ?S <= 0 |- _ ] => destruct S
adamc@86 892 | [ _ : length ?S' <= S ?N |- _ ] =>
adamc@86 893 generalize (eq_nat_dec (length S') (S N)); destruct 1
adamc@86 894 end; crush.
adamc@86 895 Defined.
adamc@86 896
adamc@93 897 (** 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@93 898
adamc@93 899 [[
adamc@93 900 | S n' => fun _ => let n := S n' in
adamc@93 901 (P1_dec (substring 0 n s)
adamc@93 902 && P2_dec (substring n (length s - n) s))
adamc@93 903 || F n' _
adamc@214 904
adamc@93 905 ]]
adamc@93 906
adamc@93 907 [split] itself is trivial to implement in terms of [split']. We just ask [split'] to begin its search with [n = length s]. *)
adamc@93 908
adamc@86 909 Definition split : {exists s1, exists s2, s = s1 ++ s2 /\ P1 s1 /\ P2 s2}
adamc@214 910 + {forall s1 s2, s = s1 ++ s2 -> ~ P1 s1 \/ ~ P2 s2}.
adamc@86 911 refine (Reduce (split' (n := length s) _)); crush; eauto.
adamc@86 912 Defined.
adamc@86 913 End split.
adamc@86 914
adamc@86 915 Implicit Arguments split [P1 P2].
adamc@86 916
adamc@93 917 (* begin hide *)
adamc@91 918 Lemma app_empty_end : forall s, s ++ "" = s.
adamc@91 919 induction s; crush.
adamc@91 920 Qed.
adamc@91 921
adamc@91 922 Hint Rewrite app_empty_end : cpdt.
adamc@91 923
adamc@91 924 Lemma substring_self : forall s n,
adamc@91 925 n <= 0
adamc@91 926 -> substring n (length s - n) s = s.
adamc@91 927 induction s; substring.
adamc@91 928 Qed.
adamc@91 929
adamc@91 930 Lemma substring_empty : forall s n m,
adamc@91 931 m <= 0
adamc@91 932 -> substring n m s = "".
adamc@91 933 induction s; substring.
adamc@91 934 Qed.
adamc@91 935
adamc@91 936 Hint Rewrite substring_self substring_empty using omega : cpdt.
adamc@91 937
adamc@91 938 Lemma substring_split' : forall s n m,
adamc@91 939 substring n m s ++ substring (n + m) (length s - (n + m)) s
adamc@91 940 = substring n (length s - n) s.
adamc@91 941 Hint Rewrite substring_split : cpdt.
adamc@91 942
adamc@91 943 induction s; substring.
adamc@91 944 Qed.
adamc@91 945
adamc@91 946 Lemma substring_stack : forall s n2 m1 m2,
adamc@91 947 m1 <= m2
adamc@91 948 -> substring 0 m1 (substring n2 m2 s)
adamc@91 949 = substring n2 m1 s.
adamc@91 950 induction s; substring.
adamc@91 951 Qed.
adamc@91 952
adamc@91 953 Ltac substring' :=
adamc@91 954 crush;
adamc@91 955 repeat match goal with
adamc@91 956 | [ |- context[match ?N with O => _ | S _ => _ end] ] => case_eq N; crush
adamc@91 957 end.
adamc@91 958
adamc@91 959 Lemma substring_stack' : forall s n1 n2 m1 m2,
adamc@91 960 n1 + m1 <= m2
adamc@91 961 -> substring n1 m1 (substring n2 m2 s)
adamc@91 962 = substring (n1 + n2) m1 s.
adamc@91 963 induction s; substring';
adamc@91 964 match goal with
adamc@91 965 | [ |- substring ?N1 _ _ = substring ?N2 _ _ ] =>
adamc@91 966 replace N1 with N2; crush
adamc@91 967 end.
adamc@91 968 Qed.
adamc@91 969
adamc@91 970 Lemma substring_suffix : forall s n,
adamc@91 971 n <= length s
adamc@91 972 -> length (substring n (length s - n) s) = length s - n.
adamc@91 973 induction s; substring.
adamc@91 974 Qed.
adamc@91 975
adamc@91 976 Lemma substring_suffix_emp' : forall s n m,
adamc@91 977 substring n (S m) s = ""
adamc@91 978 -> n >= length s.
adamc@91 979 induction s; crush;
adamc@91 980 match goal with
adamc@91 981 | [ |- ?N >= _ ] => destruct N; crush
adamc@91 982 end;
adamc@91 983 match goal with
adamc@91 984 [ |- S ?N >= S ?E ] => assert (N >= E); [ eauto | omega ]
adamc@91 985 end.
adamc@91 986 Qed.
adamc@91 987
adamc@91 988 Lemma substring_suffix_emp : forall s n m,
adamc@92 989 substring n m s = ""
adamc@92 990 -> m > 0
adamc@91 991 -> n >= length s.
adamc@91 992 destruct m as [| m]; [crush | intros; apply substring_suffix_emp' with m; assumption].
adamc@91 993 Qed.
adamc@91 994
adamc@91 995 Hint Rewrite substring_stack substring_stack' substring_suffix
adamc@91 996 using omega : cpdt.
adamc@91 997
adamc@91 998 Lemma minus_minus : forall n m1 m2,
adamc@91 999 m1 + m2 <= n
adamc@91 1000 -> n - m1 - m2 = n - (m1 + m2).
adamc@91 1001 intros; omega.
adamc@91 1002 Qed.
adamc@91 1003
adamc@91 1004 Lemma plus_n_Sm' : forall n m : nat, S (n + m) = m + S n.
adamc@91 1005 intros; omega.
adamc@91 1006 Qed.
adamc@91 1007
adamc@91 1008 Hint Rewrite minus_minus using omega : cpdt.
adamc@93 1009 (* end hide *)
adamc@93 1010
adamc@93 1011 (** 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 1012
adamc@91 1013 Section dec_star.
adamc@91 1014 Variable P : string -> Prop.
adamc@214 1015 Variable P_dec : forall s, {P s} + {~ P s}.
adamc@91 1016
adamc@93 1017 (** Some new lemmas and hints about the [star] type family are useful here. We omit them here; they are included in the book source at this point. *)
adamc@93 1018
adamc@93 1019 (* begin hide *)
adamc@91 1020 Hint Constructors star.
adamc@91 1021
adamc@91 1022 Lemma star_empty : forall s,
adamc@91 1023 length s = 0
adamc@91 1024 -> star P s.
adamc@91 1025 destruct s; crush.
adamc@91 1026 Qed.
adamc@91 1027
adamc@91 1028 Lemma star_singleton : forall s, P s -> star P s.
adamc@91 1029 intros; rewrite <- (app_empty_end s); auto.
adamc@91 1030 Qed.
adamc@91 1031
adamc@91 1032 Lemma star_app : forall s n m,
adamc@91 1033 P (substring n m s)
adamc@91 1034 -> star P (substring (n + m) (length s - (n + m)) s)
adamc@91 1035 -> star P (substring n (length s - n) s).
adamc@91 1036 induction n; substring;
adamc@91 1037 match goal with
adamc@91 1038 | [ H : P (substring ?N ?M ?S) |- _ ] =>
adamc@91 1039 solve [ rewrite <- (substring_split S M); auto
adamc@91 1040 | rewrite <- (substring_split' S N M); auto ]
adamc@91 1041 end.
adamc@91 1042 Qed.
adamc@91 1043
adamc@91 1044 Hint Resolve star_empty star_singleton star_app.
adamc@91 1045
adamc@91 1046 Variable s : string.
adamc@91 1047
adamc@91 1048 Lemma star_inv : forall s,
adamc@91 1049 star P s
adamc@91 1050 -> s = ""
adamc@91 1051 \/ exists i, i < length s
adamc@91 1052 /\ P (substring 0 (S i) s)
adamc@91 1053 /\ star P (substring (S i) (length s - S i) s).
adamc@91 1054 Hint Extern 1 (exists i : nat, _) =>
adamc@91 1055 match goal with
adamc@91 1056 | [ H : P (String _ ?S) |- _ ] => exists (length S); crush
adamc@91 1057 end.
adamc@91 1058
adamc@91 1059 induction 1; [
adamc@91 1060 crush
adamc@91 1061 | match goal with
adamc@91 1062 | [ _ : P ?S |- _ ] => destruct S; crush
adamc@91 1063 end
adamc@91 1064 ].
adamc@91 1065 Qed.
adamc@91 1066
adamc@91 1067 Lemma star_substring_inv : forall n,
adamc@91 1068 n <= length s
adamc@91 1069 -> star P (substring n (length s - n) s)
adamc@91 1070 -> substring n (length s - n) s = ""
adamc@91 1071 \/ exists l, l < length s - n
adamc@91 1072 /\ P (substring n (S l) s)
adamc@91 1073 /\ star P (substring (n + S l) (length s - (n + S l)) s).
adamc@91 1074 Hint Rewrite plus_n_Sm' : cpdt.
adamc@91 1075
adamc@91 1076 intros;
adamc@91 1077 match goal with
adamc@91 1078 | [ H : star _ _ |- _ ] => generalize (star_inv H); do 3 crush; eauto
adamc@91 1079 end.
adamc@91 1080 Qed.
adamc@93 1081 (* end hide *)
adamc@93 1082
adamc@93 1083 (** 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 1084
adamc@91 1085 Section dec_star''.
adamc@91 1086 Variable n : nat.
adamc@93 1087 (** [n] is the length of the prefix of [s] that we have already processed. *)
adamc@91 1088
adamc@91 1089 Variable P' : string -> Prop.
adamc@91 1090 Variable P'_dec : forall n' : nat, n' > n
adamc@91 1091 -> {P' (substring n' (length s - n') s)}
adamc@214 1092 + {~ P' (substring n' (length s - n') s)}.
adamc@93 1093 (** 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 1094
adamc@93 1095 (** 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 1096
adamc@91 1097 Definition dec_star'' (l : nat)
adamc@91 1098 : {exists l', S l' <= l
adamc@91 1099 /\ P (substring n (S l') s) /\ P' (substring (n + S l') (length s - (n + S l')) s)}
adamc@91 1100 + {forall l', S l' <= l
adamc@214 1101 -> ~ P (substring n (S l') s)
adamc@214 1102 \/ ~ P' (substring (n + S l') (length s - (n + S l')) s)}.
adamc@91 1103 refine (fix F (l : nat) : {exists l', S l' <= l
adamc@91 1104 /\ P (substring n (S l') s) /\ P' (substring (n + S l') (length s - (n + S l')) s)}
adamc@91 1105 + {forall l', S l' <= l
adamc@214 1106 -> ~ P (substring n (S l') s)
adamc@214 1107 \/ ~ P' (substring (n + S l') (length s - (n + S l')) s)} :=
adamc@214 1108 match l with
adamc@91 1109 | O => _
adamc@91 1110 | S l' =>
adamc@91 1111 (P_dec (substring n (S l') s) && P'_dec (n' := n + S l') _)
adamc@91 1112 || F l'
adamc@91 1113 end); clear F; crush; eauto 7;
adamc@91 1114 match goal with
adamc@91 1115 | [ H : ?X <= S ?Y |- _ ] => destruct (eq_nat_dec X (S Y)); crush
adamc@91 1116 end.
adamc@91 1117 Defined.
adamc@91 1118 End dec_star''.
adamc@91 1119
adamc@93 1120 (* begin hide *)
adamc@92 1121 Lemma star_length_contra : forall n,
adamc@92 1122 length s > n
adamc@92 1123 -> n >= length s
adamc@92 1124 -> False.
adamc@92 1125 crush.
adamc@92 1126 Qed.
adamc@92 1127
adamc@92 1128 Lemma star_length_flip : forall n n',
adamc@92 1129 length s - n <= S n'
adamc@92 1130 -> length s > n
adamc@92 1131 -> length s - n > 0.
adamc@92 1132 crush.
adamc@92 1133 Qed.
adamc@92 1134
adamc@92 1135 Hint Resolve star_length_contra star_length_flip substring_suffix_emp.
adamc@93 1136 (* end hide *)
adamc@92 1137
adamc@93 1138 (** 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 1139
adamc@91 1140 Definition dec_star' (n n' : nat) : length s - n' <= n
adamc@91 1141 -> {star P (substring n' (length s - n') s)}
adamc@214 1142 + {~ star P (substring n' (length s - n') s)}.
adamc@214 1143 refine (fix F (n n' : nat) : length s - n' <= n
adamc@91 1144 -> {star P (substring n' (length s - n') s)}
adamc@214 1145 + {~ star P (substring n' (length s - n') s)} :=
adamc@214 1146 match n with
adamc@91 1147 | O => fun _ => Yes
adamc@91 1148 | S n'' => fun _ =>
adamc@91 1149 le_gt_dec (length s) n'
adamc@91 1150 || dec_star'' (n := n') (star P) (fun n0 _ => Reduce (F n'' n0 _)) (length s - n')
adamc@92 1151 end); clear F; crush; eauto;
adamc@92 1152 match goal with
adamc@92 1153 | [ H : star _ _ |- _ ] => apply star_substring_inv in H; crush; eauto
adamc@92 1154 end;
adamc@92 1155 match goal with
adamc@92 1156 | [ H1 : _ < _ - _, H2 : forall l' : nat, _ <= _ - _ -> _ |- _ ] =>
adamc@92 1157 generalize (H2 _ (lt_le_S _ _ H1)); tauto
adamc@92 1158 end.
adamc@91 1159 Defined.
adamc@91 1160
adamc@93 1161 (** Finally, we have [dec_star]. It has a straightforward implementation. We introduce a spurious match on [s] so that [simpl] will know to reduce calls to [dec_star]. The heuristic that [simpl] uses is only to unfold identifier definitions when doing so would simplify some [match] expression. *)
adamc@93 1162
adamc@214 1163 Definition dec_star : {star P s} + {~ star P s}.
adamc@204 1164 refine (match s return _ with
adamc@91 1165 | "" => Reduce (dec_star' (n := length s) 0 _)
adamc@91 1166 | _ => Reduce (dec_star' (n := length s) 0 _)
adamc@91 1167 end); crush.
adamc@91 1168 Defined.
adamc@91 1169 End dec_star.
adamc@91 1170
adamc@93 1171 (* begin hide *)
adamc@86 1172 Lemma app_cong : forall x1 y1 x2 y2,
adamc@86 1173 x1 = x2
adamc@86 1174 -> y1 = y2
adamc@86 1175 -> x1 ++ y1 = x2 ++ y2.
adamc@86 1176 congruence.
adamc@86 1177 Qed.
adamc@86 1178
adamc@86 1179 Hint Resolve app_cong.
adamc@93 1180 (* end hide *)
adamc@93 1181
adamc@93 1182 (** 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 1183
adamc@214 1184 Definition matches P (r : regexp P) s : {P s} + {~ P s}.
adamc@214 1185 refine (fix F P (r : regexp P) s : {P s} + {~ P s} :=
adamc@86 1186 match r with
adamc@86 1187 | Char ch => string_dec s (String ch "")
adamc@86 1188 | Concat _ _ r1 r2 => Reduce (split (F _ r1) (F _ r2) s)
adamc@87 1189 | Or _ _ r1 r2 => F _ r1 s || F _ r2 s
adamc@91 1190 | Star _ r => dec_star _ _ _
adamc@86 1191 end); crush;
adamc@86 1192 match goal with
adamc@86 1193 | [ H : _ |- _ ] => generalize (H _ _ (refl_equal _))
adamc@93 1194 end; tauto.
adamc@86 1195 Defined.
adamc@86 1196
adam@283 1197 (** 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 1198
adamc@93 1199 (* begin hide *)
adamc@86 1200 Example hi := Concat (Char "h"%char) (Char "i"%char).
adamc@86 1201 Eval simpl in matches hi "hi".
adamc@86 1202 Eval simpl in matches hi "bye".
adamc@87 1203
adamc@87 1204 Example a_b := Or (Char "a"%char) (Char "b"%char).
adamc@87 1205 Eval simpl in matches a_b "".
adamc@87 1206 Eval simpl in matches a_b "a".
adamc@87 1207 Eval simpl in matches a_b "aa".
adamc@87 1208 Eval simpl in matches a_b "b".
adam@283 1209 (* end hide *)
adam@283 1210
adam@283 1211 (** 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. *)
adamc@91 1212
adamc@91 1213 Example a_star := Star (Char "a"%char).
adamc@91 1214 Eval simpl in matches a_star "".
adamc@91 1215 Eval simpl in matches a_star "a".
adamc@91 1216 Eval simpl in matches a_star "b".
adamc@91 1217 Eval simpl in matches a_star "aa".
adam@283 1218
adam@283 1219 (** 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. *)
adamc@101 1220
adamc@101 1221
adamc@101 1222 (** * Exercises *)
adamc@101 1223
adamc@101 1224 (** %\begin{enumerate}%#<ol>#
adamc@101 1225
adamc@101 1226 %\item%#<li># Define a kind of dependently-typed lists, where a list's type index gives a lower bound on how many of its elements satisfy a particular predicate. In particular, for an arbitrary set [A] and a predicate [P] over it:
adamc@101 1227 %\begin{enumerate}%#<ol>#
adamc@101 1228 %\item%#<li># Define a type [plist : nat -> Set]. Each [plist n] should be a list of [A]s, where it is guaranteed that at least [n] distinct elements satisfy [P]. There is wide latitude in choosing how to encode this. You should try to avoid using subset types or any other mechanism based on annotating non-dependent types with propositions after-the-fact.#</li>#
adamc@102 1229 %\item%#<li># Define a version of list concatenation that works on [plist]s. The type of this new function should express as much information as possible about the output [plist].#</li>#
adamc@101 1230 %\item%#<li># Define a function [plistOut] for translating [plist]s to normal [list]s.#</li>#
adamc@101 1231 %\item%#<li># Define a function [plistIn] for translating [list]s to [plist]s. The type of [plistIn] should make it clear that the best bound on [P]-matching elements is chosen. You may assume that you are given a dependently-typed function for deciding instances of [P].#</li>#
adamc@101 1232 %\item%#<li># Prove that, for any list [ls], [plistOut (plistIn ls) = ls]. This should be the only part of the exercise where you use tactic-based proving.#</li>#
adamc@101 1233 %\item%#<li># Define a function [grab : forall n (ls : plist (S n)), sig P]. That is, when given a [plist] guaranteed to contain at least one element satisfying [P], [grab] produces such an element. [sig] is the type family of sigma types, and [sig P] is extensionally equivalent to [{x : A | P x}], though the latter form uses an eta-expansion of [P] instead of [P] itself as the predicate.#</li>#
adamc@101 1234 #</ol>#%\end{enumerate}% #</li>#
adamc@101 1235
adamc@102 1236 #</ol>#%\end{enumerate}% *)