Mercurial > cpdt > repo

annotate src/MoreDep.v @ 91:4a57a4922af5

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
Add star to regexp matcher; need to automate a bit more
author Adam Chlipala <adamc@hcoop.net>
date Tue, 07 Oct 2008 14:52:15 -0400
parents 939add5a7db9
children 41392e5acbf5
rev   line source
adamc@83 1 (* Copyright (c) 2008, 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
adamc@83 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@84 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@84 43 [[
adamc@84 44 Fixpoint app n1 (ls1 : ilist n1) n2 (ls2 : ilist n2) {struct ls1} : ilist (n1 + n2) :=
adamc@84 45 match ls1 with
adamc@84 46 | Nil => ls2
adamc@84 47 | Cons _ x ls1' => Cons x (app ls1' ls2)
adamc@84 48 end.
adamc@84 49
adamc@84 50 Coq is not happy with this definition:
adamc@84 51
adamc@84 52 [[
adamc@84 53 The term "ls2" has type "ilist n2" while it is expected to have type
adamc@84 54 "ilist (?14 + n2)"
adamc@84 55 ]]
adamc@84 56
adamc@84 57 We see the return of a problem we have considered before. 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. *)
adamc@84 58
adamc@84 59 Fixpoint app n1 (ls1 : ilist n1) n2 (ls2 : ilist n2) {struct ls1} : ilist (n1 + n2) :=
adamc@84 60 match ls1 in (ilist n1) return (ilist (n1 + n2)) with
adamc@84 61 | Nil => ls2
adamc@84 62 | Cons _ x ls1' => Cons x (app ls1' ls2)
adamc@84 63 end.
adamc@84 64
adamc@84 65 (** This version of [app] passes the type checker. 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 66
adamc@84 67 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 68
adamc@84 69 Our [app] function could be typed in so-called %\textit{%#<i>#stratified#</i>#%}% type systems, which avoid true dependency. We could consider the length indices to lists to live in a separate, compile-time-only universe from the lists themselves. 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 70
adamc@84 71 Fixpoint inject (ls : list A) : ilist (length ls) :=
adamc@84 72 match ls return (ilist (length ls)) with
adamc@84 73 | nil => Nil
adamc@84 74 | h :: t => Cons h (inject t)
adamc@84 75 end.
adamc@84 76
adamc@84 77 (** We can define an inverse conversion and prove that it really is an inverse. *)
adamc@84 78
adamc@84 79 Fixpoint unject n (ls : ilist n) {struct ls} : list A :=
adamc@84 80 match ls with
adamc@84 81 | Nil => nil
adamc@84 82 | Cons _ h t => h :: unject t
adamc@84 83 end.
adamc@84 84
adamc@84 85 Theorem inject_inverse : forall ls, unject (inject ls) = ls.
adamc@84 86 induction ls; crush.
adamc@84 87 Qed.
adamc@84 88
adamc@84 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.
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@84 97
adamc@84 98 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 99
adamc@84 100 [[
adamc@84 101 Definition hd n (ls : ilist (S n)) : A :=
adamc@84 102 match ls with
adamc@84 103 | Cons _ h _ => h
adamc@84 104 end.
adamc@84 105
adamc@84 106 [[
adamc@84 107 Error: Non exhaustive pattern-matching: no clause found for pattern Nil
adamc@84 108 ]]
adamc@84 109
adamc@84 110 Unlike in ML, we cannot use inexhaustive pattern matching, becuase there is no conception of a %\texttt{%#<tt>#Match#</tt>#%}% exception to be thrown. We might try using an [in] clause somehow.
adamc@84 111
adamc@84 112 [[
adamc@84 113 Definition hd n (ls : ilist (S n)) : A :=
adamc@84 114 match ls in (ilist (S n)) with
adamc@84 115 | Cons _ h _ => h
adamc@84 116 end.
adamc@84 117
adamc@84 118 [[
adamc@84 119 Error: The reference n was not found in the current environment
adamc@84 120 ]]
adamc@84 121
adamc@84 122 In this and other cases, we feel like we want [in] clauses with type family arguments that are not variables. Unfortunately, Coq only supports variables in those positions. A completely general mechanism could only be supported with a solution to the problem of higher-order unification, 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@84 123
adamc@84 124 Our final, working attempt at [hd] uses an auxiliary function and a surprising [return] annotation. *)
adamc@84 125
adamc@84 126 Definition hd' n (ls : ilist n) :=
adamc@84 127 match ls in (ilist n) return (match n with O => unit | S _ => A end) with
adamc@84 128 | Nil => tt
adamc@84 129 | Cons _ h _ => h
adamc@84 130 end.
adamc@84 131
adamc@84 132 Definition hd n (ls : ilist (S n)) : A := hd' ls.
adamc@84 133
adamc@84 134 (** 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@84 135
adamc@84 136 End ilist.
adamc@85 137
adamc@85 138
adamc@85 139 (** * A Tagless Interpreter *)
adamc@85 140
adamc@85 141 (** 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@85 142
adamc@85 143 Inductive type : Set :=
adamc@85 144 | Nat : type
adamc@85 145 | Bool : type
adamc@85 146 | Prod : type -> type -> type.
adamc@85 147
adamc@85 148 Inductive exp : type -> Set :=
adamc@85 149 | NConst : nat -> exp Nat
adamc@85 150 | Plus : exp Nat -> exp Nat -> exp Nat
adamc@85 151 | Eq : exp Nat -> exp Nat -> exp Bool
adamc@85 152
adamc@85 153 | BConst : bool -> exp Bool
adamc@85 154 | And : exp Bool -> exp Bool -> exp Bool
adamc@85 155 | If : forall t, exp Bool -> exp t -> exp t -> exp t
adamc@85 156
adamc@85 157 | Pair : forall t1 t2, exp t1 -> exp t2 -> exp (Prod t1 t2)
adamc@85 158 | Fst : forall t1 t2, exp (Prod t1 t2) -> exp t1
adamc@85 159 | Snd : forall t1 t2, exp (Prod t1 t2) -> exp t2.
adamc@85 160
adamc@85 161 (** 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@85 162
adamc@85 163 We can give types and expressions semantics in a new style, based critically on the chance for %\textit{%#<i>#type-level computation#</i>#%}%. *)
adamc@85 164
adamc@85 165 Fixpoint typeDenote (t : type) : Set :=
adamc@85 166 match t with
adamc@85 167 | Nat => nat
adamc@85 168 | Bool => bool
adamc@85 169 | Prod t1 t2 => typeDenote t1 * typeDenote t2
adamc@85 170 end%type.
adamc@85 171
adamc@85 172 (** [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@85 173
adamc@85 174 We can define a function [expDenote] that is typed in terms of [typeDenote]. *)
adamc@85 175
adamc@85 176 Fixpoint expDenote t (e : exp t) {struct e} : typeDenote t :=
adamc@85 177 match e in (exp t) return (typeDenote t) with
adamc@85 178 | NConst n => n
adamc@85 179 | Plus e1 e2 => expDenote e1 + expDenote e2
adamc@85 180 | Eq e1 e2 => if eq_nat_dec (expDenote e1) (expDenote e2) then true else false
adamc@85 181
adamc@85 182 | BConst b => b
adamc@85 183 | And e1 e2 => expDenote e1 && expDenote e2
adamc@85 184 | If _ e' e1 e2 => if expDenote e' then expDenote e1 else expDenote e2
adamc@85 185
adamc@85 186 | Pair _ _ e1 e2 => (expDenote e1, expDenote e2)
adamc@85 187 | Fst _ _ e' => fst (expDenote e')
adamc@85 188 | Snd _ _ e' => snd (expDenote e')
adamc@85 189 end.
adamc@85 190
adamc@85 191 (** Again we find that an [in] annotation is essential for type-checking a function. Besides that, the 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@85 192
adamc@85 193 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@85 194
adamc@85 195 [[
adamc@85 196 Definition pairOut t1 t2 (e : exp (Prod t1 t2)) : option (exp t1 * exp t2) :=
adamc@85 197 match e in (exp (Prod t1 t2)) return option (exp t1 * exp t2) with
adamc@85 198 | Pair _ _ e1 e2 => Some (e1, e2)
adamc@85 199 | _ => None
adamc@85 200 end.
adamc@85 201
adamc@85 202 [[
adamc@85 203 Error: The reference t2 was not found in the current environment
adamc@85 204 ]]
adamc@85 205
adamc@85 206 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@85 207
adamc@85 208 Definition pairOutType (t : type) :=
adamc@85 209 match t with
adamc@85 210 | Prod t1 t2 => option (exp t1 * exp t2)
adamc@85 211 | _ => unit
adamc@85 212 end.
adamc@85 213
adamc@85 214 (** 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@85 215
adamc@85 216 Definition pairOutDefault (t : type) :=
adamc@85 217 match t return (pairOutType t) with
adamc@85 218 | Prod _ _ => None
adamc@85 219 | _ => tt
adamc@85 220 end.
adamc@85 221
adamc@85 222 (** Now [pairOut] is deceptively easy to write. *)
adamc@85 223
adamc@85 224 Definition pairOut t (e : exp t) :=
adamc@85 225 match e in (exp t) return (pairOutType t) with
adamc@85 226 | Pair _ _ e1 e2 => Some (e1, e2)
adamc@85 227 | _ => pairOutDefault _
adamc@85 228 end.
adamc@85 229
adamc@85 230 (** 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@85 231
adamc@85 232 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. *)
adamc@85 233
adamc@85 234 Fixpoint cfold t (e : exp t) {struct e} : exp t :=
adamc@85 235 match e in (exp t) return (exp t) with
adamc@85 236 | NConst n => NConst n
adamc@85 237 | Plus e1 e2 =>
adamc@85 238 let e1' := cfold e1 in
adamc@85 239 let e2' := cfold e2 in
adamc@85 240 match e1', e2' with
adamc@85 241 | NConst n1, NConst n2 => NConst (n1 + n2)
adamc@85 242 | _, _ => Plus e1' e2'
adamc@85 243 end
adamc@85 244 | Eq e1 e2 =>
adamc@85 245 let e1' := cfold e1 in
adamc@85 246 let e2' := cfold e2 in
adamc@85 247 match e1', e2' with
adamc@85 248 | NConst n1, NConst n2 => BConst (if eq_nat_dec n1 n2 then true else false)
adamc@85 249 | _, _ => Eq e1' e2'
adamc@85 250 end
adamc@85 251
adamc@85 252 | BConst b => BConst b
adamc@85 253 | And e1 e2 =>
adamc@85 254 let e1' := cfold e1 in
adamc@85 255 let e2' := cfold e2 in
adamc@85 256 match e1', e2' with
adamc@85 257 | BConst b1, BConst b2 => BConst (b1 && b2)
adamc@85 258 | _, _ => And e1' e2'
adamc@85 259 end
adamc@85 260 | If _ e e1 e2 =>
adamc@85 261 let e' := cfold e in
adamc@85 262 match e' with
adamc@85 263 | BConst true => cfold e1
adamc@85 264 | BConst false => cfold e2
adamc@85 265 | _ => If e' (cfold e1) (cfold e2)
adamc@85 266 end
adamc@85 267
adamc@85 268 | Pair _ _ e1 e2 => Pair (cfold e1) (cfold e2)
adamc@85 269 | Fst _ _ e =>
adamc@85 270 let e' := cfold e in
adamc@85 271 match pairOut e' with
adamc@85 272 | Some p => fst p
adamc@85 273 | None => Fst e'
adamc@85 274 end
adamc@85 275 | Snd _ _ e =>
adamc@85 276 let e' := cfold e in
adamc@85 277 match pairOut e' with
adamc@85 278 | Some p => snd p
adamc@85 279 | None => Snd e'
adamc@85 280 end
adamc@85 281 end.
adamc@85 282
adamc@85 283 (** The correctness theorem for [cfold] turns out to be easy to prove, once we get over one serious hurdle. *)
adamc@85 284
adamc@85 285 Theorem cfold_correct : forall t (e : exp t), expDenote e = expDenote (cfold e).
adamc@85 286 induction e; crush.
adamc@85 287
adamc@85 288 (** The first remaining subgoal is:
adamc@85 289
adamc@85 290 [[
adamc@85 291
adamc@85 292 expDenote (cfold e1) + expDenote (cfold e2) =
adamc@85 293 expDenote
adamc@85 294 match cfold e1 with
adamc@85 295 | NConst n1 =>
adamc@85 296 match cfold e2 with
adamc@85 297 | NConst n2 => NConst (n1 + n2)
adamc@85 298 | Plus _ _ => Plus (cfold e1) (cfold e2)
adamc@85 299 | Eq _ _ => Plus (cfold e1) (cfold e2)
adamc@85 300 | BConst _ => Plus (cfold e1) (cfold e2)
adamc@85 301 | And _ _ => Plus (cfold e1) (cfold e2)
adamc@85 302 | If _ _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 303 | Pair _ _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 304 | Fst _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 305 | Snd _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 306 end
adamc@85 307 | Plus _ _ => Plus (cfold e1) (cfold e2)
adamc@85 308 | Eq _ _ => Plus (cfold e1) (cfold e2)
adamc@85 309 | BConst _ => Plus (cfold e1) (cfold e2)
adamc@85 310 | And _ _ => Plus (cfold e1) (cfold e2)
adamc@85 311 | If _ _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 312 | Pair _ _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 313 | Fst _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 314 | Snd _ _ _ => Plus (cfold e1) (cfold e2)
adamc@85 315 end
adamc@85 316 ]]
adamc@85 317
adamc@85 318 We would like to do a case analysis on [cfold e1], and we attempt that in the way that has worked so far.
adamc@85 319
adamc@85 320 [[
adamc@85 321 destruct (cfold e1).
adamc@85 322
adamc@85 323 [[
adamc@85 324 User error: e1 is used in hypothesis e
adamc@85 325 ]]
adamc@85 326
adamc@85 327 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@85 328
adamc@85 329 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. 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@85 330
adamc@85 331 dep_destruct (cfold e1).
adamc@85 332
adamc@85 333 (** 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@85 334
adamc@85 335 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@85 336
adamc@85 337 Restart.
adamc@85 338
adamc@85 339 induction e; crush;
adamc@85 340 repeat (match goal with
adamc@85 341 | [ |- context[cfold ?E] ] => dep_destruct (cfold E)
adamc@85 342 | [ |- (if ?E then _ else _) = _ ] => destruct E
adamc@85 343 end; crush).
adamc@85 344 Qed.
adamc@86 345
adamc@86 346
adamc@86 347 (** * A Certified Regular Expression Matcher *)
adamc@86 348
adamc@86 349 Require Import Ascii String.
adamc@86 350 Open Scope string_scope.
adamc@86 351
adamc@91 352 Section star.
adamc@91 353 Variable P : string -> Prop.
adamc@91 354
adamc@91 355 Inductive star : string -> Prop :=
adamc@91 356 | Empty : star ""
adamc@91 357 | Iter : forall s1 s2,
adamc@91 358 P s1
adamc@91 359 -> star s2
adamc@91 360 -> star (s1 ++ s2).
adamc@91 361 End star.
adamc@91 362
adamc@89 363 Inductive regexp : (string -> Prop) -> Type :=
adamc@86 364 | Char : forall ch : ascii,
adamc@86 365 regexp (fun s => s = String ch "")
adamc@86 366 | Concat : forall P1 P2 (r1 : regexp P1) (r2 : regexp P2),
adamc@87 367 regexp (fun s => exists s1, exists s2, s = s1 ++ s2 /\ P1 s1 /\ P2 s2)
adamc@87 368 | Or : forall P1 P2 (r1 : regexp P1) (r2 : regexp P2),
adamc@91 369 regexp (fun s => P1 s \/ P2 s)
adamc@91 370 | Star : forall P (r : regexp P),
adamc@91 371 regexp (star P).
adamc@86 372
adamc@86 373 Open Scope specif_scope.
adamc@86 374
adamc@86 375 Lemma length_emp : length "" <= 0.
adamc@86 376 crush.
adamc@86 377 Qed.
adamc@86 378
adamc@86 379 Lemma append_emp : forall s, s = "" ++ s.
adamc@86 380 crush.
adamc@86 381 Qed.
adamc@86 382
adamc@86 383 Ltac substring :=
adamc@86 384 crush;
adamc@86 385 repeat match goal with
adamc@86 386 | [ |- context[match ?N with O => _ | S _ => _ end] ] => destruct N; crush
adamc@86 387 end.
adamc@86 388
adamc@86 389 Lemma substring_le : forall s n m,
adamc@86 390 length (substring n m s) <= m.
adamc@86 391 induction s; substring.
adamc@86 392 Qed.
adamc@86 393
adamc@86 394 Lemma substring_all : forall s,
adamc@86 395 substring 0 (length s) s = s.
adamc@86 396 induction s; substring.
adamc@86 397 Qed.
adamc@86 398
adamc@86 399 Lemma substring_none : forall s n,
adamc@86 400 substring n 0 s = EmptyString.
adamc@86 401 induction s; substring.
adamc@86 402 Qed.
adamc@86 403
adamc@86 404 Hint Rewrite substring_all substring_none : cpdt.
adamc@86 405
adamc@86 406 Lemma substring_split : forall s m,
adamc@86 407 substring 0 m s ++ substring m (length s - m) s = s.
adamc@86 408 induction s; substring.
adamc@86 409 Qed.
adamc@86 410
adamc@86 411 Lemma length_app1 : forall s1 s2,
adamc@86 412 length s1 <= length (s1 ++ s2).
adamc@86 413 induction s1; crush.
adamc@86 414 Qed.
adamc@86 415
adamc@86 416 Hint Resolve length_emp append_emp substring_le substring_split length_app1.
adamc@86 417
adamc@86 418 Lemma substring_app_fst : forall s2 s1 n,
adamc@86 419 length s1 = n
adamc@86 420 -> substring 0 n (s1 ++ s2) = s1.
adamc@86 421 induction s1; crush.
adamc@86 422 Qed.
adamc@86 423
adamc@86 424 Lemma substring_app_snd : forall s2 s1 n,
adamc@86 425 length s1 = n
adamc@86 426 -> substring n (length (s1 ++ s2) - n) (s1 ++ s2) = s2.
adamc@86 427 Hint Rewrite <- minus_n_O : cpdt.
adamc@86 428
adamc@86 429 induction s1; crush.
adamc@86 430 Qed.
adamc@86 431
adamc@91 432 Hint Rewrite substring_app_fst substring_app_snd using (trivial; fail) : cpdt.
adamc@86 433
adamc@86 434 Section split.
adamc@86 435 Variables P1 P2 : string -> Prop.
adamc@91 436 Variable P1_dec : forall s, {P1 s} + { ~P1 s}.
adamc@91 437 Variable P2_dec : forall s, {P2 s} + { ~P2 s}.
adamc@86 438
adamc@86 439 Variable s : string.
adamc@86 440
adamc@86 441 Definition split' (n : nat) : n <= length s
adamc@86 442 -> {exists s1, exists s2, length s1 <= n /\ s1 ++ s2 = s /\ P1 s1 /\ P2 s2}
adamc@86 443 + {forall s1 s2, length s1 <= n -> s1 ++ s2 = s -> ~P1 s1 \/ ~P2 s2}.
adamc@86 444 refine (fix F (n : nat) : n <= length s
adamc@86 445 -> {exists s1, exists s2, length s1 <= n /\ s1 ++ s2 = s /\ P1 s1 /\ P2 s2}
adamc@86 446 + {forall s1 s2, length s1 <= n -> s1 ++ s2 = s -> ~P1 s1 \/ ~P2 s2} :=
adamc@86 447 match n return n <= length s
adamc@86 448 -> {exists s1, exists s2, length s1 <= n /\ s1 ++ s2 = s /\ P1 s1 /\ P2 s2}
adamc@86 449 + {forall s1 s2, length s1 <= n -> s1 ++ s2 = s -> ~P1 s1 \/ ~P2 s2} with
adamc@86 450 | O => fun _ => Reduce (P1_dec "" && P2_dec s)
adamc@86 451 | S n' => fun _ => (P1_dec (substring 0 (S n') s) && P2_dec (substring (S n') (length s - S n') s))
adamc@86 452 || F n' _
adamc@86 453 end); clear F; crush; eauto 7;
adamc@86 454 match goal with
adamc@86 455 | [ _ : length ?S <= 0 |- _ ] => destruct S
adamc@86 456 | [ _ : length ?S' <= S ?N |- _ ] =>
adamc@86 457 generalize (eq_nat_dec (length S') (S N)); destruct 1
adamc@86 458 end; crush.
adamc@86 459 Defined.
adamc@86 460
adamc@86 461 Definition split : {exists s1, exists s2, s = s1 ++ s2 /\ P1 s1 /\ P2 s2}
adamc@86 462 + {forall s1 s2, s = s1 ++ s2 -> ~P1 s1 \/ ~P2 s2}.
adamc@86 463 refine (Reduce (split' (n := length s) _)); crush; eauto.
adamc@86 464 Defined.
adamc@86 465 End split.
adamc@86 466
adamc@86 467 Implicit Arguments split [P1 P2].
adamc@86 468
adamc@91 469 Lemma app_empty_end : forall s, s ++ "" = s.
adamc@91 470 induction s; crush.
adamc@91 471 Qed.
adamc@91 472
adamc@91 473 Hint Rewrite app_empty_end : cpdt.
adamc@91 474
adamc@91 475 Lemma substring_self : forall s n,
adamc@91 476 n <= 0
adamc@91 477 -> substring n (length s - n) s = s.
adamc@91 478 induction s; substring.
adamc@91 479 Qed.
adamc@91 480
adamc@91 481 Lemma substring_empty : forall s n m,
adamc@91 482 m <= 0
adamc@91 483 -> substring n m s = "".
adamc@91 484 induction s; substring.
adamc@91 485 Qed.
adamc@91 486
adamc@91 487 Hint Rewrite substring_self substring_empty using omega : cpdt.
adamc@91 488
adamc@91 489 Lemma substring_split' : forall s n m,
adamc@91 490 substring n m s ++ substring (n + m) (length s - (n + m)) s
adamc@91 491 = substring n (length s - n) s.
adamc@91 492 Hint Rewrite substring_split : cpdt.
adamc@91 493
adamc@91 494 induction s; substring.
adamc@91 495 Qed.
adamc@91 496
adamc@91 497 Lemma substring_stack : forall s n2 m1 m2,
adamc@91 498 m1 <= m2
adamc@91 499 -> substring 0 m1 (substring n2 m2 s)
adamc@91 500 = substring n2 m1 s.
adamc@91 501 induction s; substring.
adamc@91 502 Qed.
adamc@91 503
adamc@91 504 Ltac substring' :=
adamc@91 505 crush;
adamc@91 506 repeat match goal with
adamc@91 507 | [ |- context[match ?N with O => _ | S _ => _ end] ] => case_eq N; crush
adamc@91 508 end.
adamc@91 509
adamc@91 510 Lemma substring_stack' : forall s n1 n2 m1 m2,
adamc@91 511 n1 + m1 <= m2
adamc@91 512 -> substring n1 m1 (substring n2 m2 s)
adamc@91 513 = substring (n1 + n2) m1 s.
adamc@91 514 induction s; substring';
adamc@91 515 match goal with
adamc@91 516 | [ |- substring ?N1 _ _ = substring ?N2 _ _ ] =>
adamc@91 517 replace N1 with N2; crush
adamc@91 518 end.
adamc@91 519 Qed.
adamc@91 520
adamc@91 521 Lemma substring_suffix : forall s n,
adamc@91 522 n <= length s
adamc@91 523 -> length (substring n (length s - n) s) = length s - n.
adamc@91 524 induction s; substring.
adamc@91 525 Qed.
adamc@91 526
adamc@91 527 Lemma substring_suffix_emp' : forall s n m,
adamc@91 528 substring n (S m) s = ""
adamc@91 529 -> n >= length s.
adamc@91 530 induction s; crush;
adamc@91 531 match goal with
adamc@91 532 | [ |- ?N >= _ ] => destruct N; crush
adamc@91 533 end;
adamc@91 534 match goal with
adamc@91 535 [ |- S ?N >= S ?E ] => assert (N >= E); [ eauto | omega ]
adamc@91 536 end.
adamc@91 537 Qed.
adamc@91 538
adamc@91 539 Lemma substring_suffix_emp : forall s n m,
adamc@91 540 m > 0
adamc@91 541 -> substring n m s = ""
adamc@91 542 -> n >= length s.
adamc@91 543 destruct m as [| m]; [crush | intros; apply substring_suffix_emp' with m; assumption].
adamc@91 544 Qed.
adamc@91 545
adamc@91 546 Hint Rewrite substring_stack substring_stack' substring_suffix
adamc@91 547 using omega : cpdt.
adamc@91 548
adamc@91 549 Lemma minus_minus : forall n m1 m2,
adamc@91 550 m1 + m2 <= n
adamc@91 551 -> n - m1 - m2 = n - (m1 + m2).
adamc@91 552 intros; omega.
adamc@91 553 Qed.
adamc@91 554
adamc@91 555 Lemma plus_n_Sm' : forall n m : nat, S (n + m) = m + S n.
adamc@91 556 intros; omega.
adamc@91 557 Qed.
adamc@91 558
adamc@91 559 Hint Rewrite minus_minus using omega : cpdt.
adamc@91 560
adamc@91 561 Section dec_star.
adamc@91 562 Variable P : string -> Prop.
adamc@91 563 Variable P_dec : forall s, {P s} + { ~P s}.
adamc@91 564
adamc@91 565 Hint Constructors star.
adamc@91 566
adamc@91 567 Lemma star_empty : forall s,
adamc@91 568 length s = 0
adamc@91 569 -> star P s.
adamc@91 570 destruct s; crush.
adamc@91 571 Qed.
adamc@91 572
adamc@91 573 Lemma star_singleton : forall s, P s -> star P s.
adamc@91 574 intros; rewrite <- (app_empty_end s); auto.
adamc@91 575 Qed.
adamc@91 576
adamc@91 577 Lemma star_app : forall s n m,
adamc@91 578 P (substring n m s)
adamc@91 579 -> star P (substring (n + m) (length s - (n + m)) s)
adamc@91 580 -> star P (substring n (length s - n) s).
adamc@91 581 induction n; substring;
adamc@91 582 match goal with
adamc@91 583 | [ H : P (substring ?N ?M ?S) |- _ ] =>
adamc@91 584 solve [ rewrite <- (substring_split S M); auto
adamc@91 585 | rewrite <- (substring_split' S N M); auto ]
adamc@91 586 end.
adamc@91 587 Qed.
adamc@91 588
adamc@91 589 Hint Resolve star_empty star_singleton star_app.
adamc@91 590
adamc@91 591 Variable s : string.
adamc@91 592
adamc@91 593 Lemma star_inv : forall s,
adamc@91 594 star P s
adamc@91 595 -> s = ""
adamc@91 596 \/ exists i, i < length s
adamc@91 597 /\ P (substring 0 (S i) s)
adamc@91 598 /\ star P (substring (S i) (length s - S i) s).
adamc@91 599 Hint Extern 1 (exists i : nat, _) =>
adamc@91 600 match goal with
adamc@91 601 | [ H : P (String _ ?S) |- _ ] => exists (length S); crush
adamc@91 602 end.
adamc@91 603
adamc@91 604 induction 1; [
adamc@91 605 crush
adamc@91 606 | match goal with
adamc@91 607 | [ _ : P ?S |- _ ] => destruct S; crush
adamc@91 608 end
adamc@91 609 ].
adamc@91 610 Qed.
adamc@91 611
adamc@91 612 Lemma star_substring_inv : forall n,
adamc@91 613 n <= length s
adamc@91 614 -> star P (substring n (length s - n) s)
adamc@91 615 -> substring n (length s - n) s = ""
adamc@91 616 \/ exists l, l < length s - n
adamc@91 617 /\ P (substring n (S l) s)
adamc@91 618 /\ star P (substring (n + S l) (length s - (n + S l)) s).
adamc@91 619 Hint Rewrite plus_n_Sm' : cpdt.
adamc@91 620
adamc@91 621 intros;
adamc@91 622 match goal with
adamc@91 623 | [ H : star _ _ |- _ ] => generalize (star_inv H); do 3 crush; eauto
adamc@91 624 end.
adamc@91 625 Qed.
adamc@91 626
adamc@91 627 Section dec_star''.
adamc@91 628 Variable n : nat.
adamc@91 629
adamc@91 630 Variable P' : string -> Prop.
adamc@91 631 Variable P'_dec : forall n' : nat, n' > n
adamc@91 632 -> {P' (substring n' (length s - n') s)}
adamc@91 633 + { ~P' (substring n' (length s - n') s)}.
adamc@91 634
adamc@91 635 Definition dec_star'' (l : nat)
adamc@91 636 : {exists l', S l' <= l
adamc@91 637 /\ P (substring n (S l') s) /\ P' (substring (n + S l') (length s - (n + S l')) s)}
adamc@91 638 + {forall l', S l' <= l
adamc@91 639 -> ~P (substring n (S l') s) \/ ~P' (substring (n + S l') (length s - (n + S l')) s)}.
adamc@91 640 refine (fix F (l : nat) : {exists l', S l' <= l
adamc@91 641 /\ P (substring n (S l') s) /\ P' (substring (n + S l') (length s - (n + S l')) s)}
adamc@91 642 + {forall l', S l' <= l
adamc@91 643 -> ~P (substring n (S l') s) \/ ~P' (substring (n + S l') (length s - (n + S l')) s)} :=
adamc@91 644 match l return {exists l', S l' <= l
adamc@91 645 /\ P (substring n (S l') s) /\ P' (substring (n + S l') (length s - (n + S l')) s)}
adamc@91 646 + {forall l', S l' <= l ->
adamc@91 647 ~P (substring n (S l') s) \/ ~P' (substring (n + S l') (length s - (n + S l')) s)} with
adamc@91 648 | O => _
adamc@91 649 | S l' =>
adamc@91 650 (P_dec (substring n (S l') s) && P'_dec (n' := n + S l') _)
adamc@91 651 || F l'
adamc@91 652 end); clear F; crush; eauto 7;
adamc@91 653 match goal with
adamc@91 654 | [ H : ?X <= S ?Y |- _ ] => destruct (eq_nat_dec X (S Y)); crush
adamc@91 655 end.
adamc@91 656 Defined.
adamc@91 657 End dec_star''.
adamc@91 658
adamc@91 659 Definition dec_star' (n n' : nat) : length s - n' <= n
adamc@91 660 -> {star P (substring n' (length s - n') s)}
adamc@91 661 + {~star P (substring n' (length s - n') s)}.
adamc@91 662 About dec_star''.
adamc@91 663
adamc@91 664 refine (fix F (n n' : nat) {struct n} : length s - n' <= n
adamc@91 665 -> {star P (substring n' (length s - n') s)}
adamc@91 666 + {~star P (substring n' (length s - n') s)} :=
adamc@91 667 match n return length s - n' <= n
adamc@91 668 -> {star P (substring n' (length s - n') s)}
adamc@91 669 + {~star P (substring n' (length s - n') s)} with
adamc@91 670 | O => fun _ => Yes
adamc@91 671 | S n'' => fun _ =>
adamc@91 672 le_gt_dec (length s) n'
adamc@91 673 || dec_star'' (n := n') (star P) (fun n0 _ => Reduce (F n'' n0 _)) (length s - n')
adamc@91 674 end); clear F; crush; eauto.
adamc@91 675
adamc@91 676 apply star_substring_inv in H; crush; eauto.
adamc@91 677
adamc@91 678 assert (n' >= length s); [ | omega].
adamc@91 679 apply substring_suffix_emp with (length s - n'); crush.
adamc@91 680
adamc@91 681 assert (S x <= length s - n'); [ omega | ].
adamc@91 682 apply _1 in H1.
adamc@91 683 tauto.
adamc@91 684 Defined.
adamc@91 685
adamc@91 686 Definition dec_star : {star P s} + { ~star P s}.
adamc@91 687 refine (match s with
adamc@91 688 | "" => Reduce (dec_star' (n := length s) 0 _)
adamc@91 689 | _ => Reduce (dec_star' (n := length s) 0 _)
adamc@91 690 end); crush.
adamc@91 691 Defined.
adamc@91 692 End dec_star.
adamc@91 693
adamc@86 694 Lemma app_cong : forall x1 y1 x2 y2,
adamc@86 695 x1 = x2
adamc@86 696 -> y1 = y2
adamc@86 697 -> x1 ++ y1 = x2 ++ y2.
adamc@86 698 congruence.
adamc@86 699 Qed.
adamc@86 700
adamc@86 701 Hint Resolve app_cong.
adamc@86 702
adamc@91 703
adamc@91 704
adamc@86 705 Definition matches P (r : regexp P) s : {P s} + { ~P s}.
adamc@86 706 refine (fix F P (r : regexp P) s : {P s} + { ~P s} :=
adamc@86 707 match r with
adamc@86 708 | Char ch => string_dec s (String ch "")
adamc@86 709 | Concat _ _ r1 r2 => Reduce (split (F _ r1) (F _ r2) s)
adamc@87 710 | Or _ _ r1 r2 => F _ r1 s || F _ r2 s
adamc@91 711 | Star _ r => dec_star _ _ _
adamc@86 712 end); crush;
adamc@86 713 match goal with
adamc@86 714 | [ H : _ |- _ ] => generalize (H _ _ (refl_equal _))
adamc@86 715 end;
adamc@86 716 tauto.
adamc@86 717 Defined.
adamc@86 718
adamc@86 719 Example hi := Concat (Char "h"%char) (Char "i"%char).
adamc@86 720 Eval simpl in matches hi "hi".
adamc@86 721 Eval simpl in matches hi "bye".
adamc@87 722
adamc@87 723 Example a_b := Or (Char "a"%char) (Char "b"%char).
adamc@87 724 Eval simpl in matches a_b "".
adamc@87 725 Eval simpl in matches a_b "a".
adamc@87 726 Eval simpl in matches a_b "aa".
adamc@87 727 Eval simpl in matches a_b "b".
adamc@91 728
adamc@91 729 Example a_star := Star (Char "a"%char).
adamc@91 730 Eval simpl in matches a_star "".
adamc@91 731 Eval simpl in matches a_star "a".
adamc@91 732 Eval simpl in matches a_star "b".
adamc@91 733 Eval simpl in matches a_star "aa".