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Pass through Chapter 13
author Adam Chlipala <adam@chlipala.net>
date Sun, 10 Feb 2013 16:00:26 -0500
parents b7419a10e52e
children 8921cfa2f503
rev   line source
adam@380 1 (* Copyright (c) 2008-2012, Adam Chlipala
adamc@70 2 *
adamc@70 3 * This work is licensed under a
adamc@70 4 * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
adamc@70 5 * Unported License.
adamc@70 6 * The license text is available at:
adamc@70 7 * http://creativecommons.org/licenses/by-nc-nd/3.0/
adamc@70 8 *)
adamc@70 9
adamc@70 10 (* begin hide *)
adamc@70 11 Require Import List.
adamc@70 12
adam@314 13 Require Import CpdtTactics.
adamc@70 14
adamc@70 15 Set Implicit Arguments.
adamc@70 16 (* end hide *)
adamc@70 17
adam@403 18 (** printing <-- $\longleftarrow$ *)
adam@403 19
adamc@70 20
adamc@74 21 (** %\part{Programming with Dependent Types}
adamc@74 22
adamc@74 23 \chapter{Subset Types and Variations}% *)
adamc@70 24
adam@423 25 (** So far, we have seen many examples of what we might call "classical program verification." We write programs, write their specifications, and then prove that the programs satisfy their specifications. The programs that we have written in Coq have been normal functional programs that we could just as well have written in Haskell or ML. In this chapter, we start investigating uses of%\index{dependent types}% _dependent types_ to integrate programming, specification, and proving into a single phase. The techniques we will learn make it possible to reduce the cost of program verification dramatically. *)
adamc@70 26
adamc@70 27
adamc@70 28 (** * Introducing Subset Types *)
adamc@70 29
adamc@70 30 (** Let us consider several ways of implementing the natural number predecessor function. We start by displaying the definition from the standard library: *)
adamc@70 31
adamc@70 32 Print pred.
adamc@212 33 (** %\vspace{-.15in}% [[
adamc@70 34 pred = fun n : nat => match n with
adamc@70 35 | 0 => 0
adamc@70 36 | S u => u
adamc@70 37 end
adamc@70 38 : nat -> nat
adamc@212 39
adamc@212 40 ]]
adamc@70 41
adam@335 42 We can use a new command, %\index{Vernacular commands!Extraction}\index{program extraction}\index{extraction|see{program extraction}}%[Extraction], to produce an %\index{OCaml}%OCaml version of this function. *)
adamc@70 43
adamc@70 44 Extraction pred.
adamc@70 45
adamc@70 46 (** %\begin{verbatim}
adamc@70 47 (** val pred : nat -> nat **)
adamc@70 48
adamc@70 49 let pred = function
adamc@70 50 | O -> O
adamc@70 51 | S u -> u
adamc@70 52 \end{verbatim}%
adamc@70 53
adamc@70 54 #<pre>
adamc@70 55 (** val pred : nat -> nat **)
adamc@70 56
adamc@70 57 let pred = function
adamc@70 58 | O -> O
adamc@70 59 | S u -> u
adamc@70 60 </pre># *)
adamc@70 61
adamc@70 62 (** Returning 0 as the predecessor of 0 can come across as somewhat of a hack. In some situations, we might like to be sure that we never try to take the predecessor of 0. We can enforce this by giving [pred] a stronger, dependent type. *)
adamc@70 63
adamc@70 64 Lemma zgtz : 0 > 0 -> False.
adamc@70 65 crush.
adamc@70 66 Qed.
adamc@70 67
adamc@70 68 Definition pred_strong1 (n : nat) : n > 0 -> nat :=
adamc@212 69 match n with
adamc@70 70 | O => fun pf : 0 > 0 => match zgtz pf with end
adamc@70 71 | S n' => fun _ => n'
adamc@70 72 end.
adamc@70 73
adam@398 74 (** We expand the type of [pred] to include a _proof_ that its argument [n] is greater than 0. When [n] is 0, we use the proof to derive a contradiction, which we can use to build a value of any type via a vacuous pattern match. When [n] is a successor, we have no need for the proof and just return the answer. The proof argument can be said to have a _dependent_ type, because its type depends on the _value_ of the argument [n].
adamc@70 75
adam@398 76 Coq's [Eval] command can execute particular invocations of [pred_strong1] just as easily as it can execute more traditional functional programs. Note that Coq has decided that argument [n] of [pred_strong1] can be made _implicit_, since it can be deduced from the type of the second argument, so we need not write [n] in function calls. *)
adam@282 77
adam@282 78 Theorem two_gt0 : 2 > 0.
adam@282 79 crush.
adam@282 80 Qed.
adam@282 81
adam@282 82 Eval compute in pred_strong1 two_gt0.
adam@282 83 (** %\vspace{-.15in}% [[
adam@282 84 = 1
adam@282 85 : nat
adam@282 86 ]]
adam@282 87
adam@442 88 One aspect in particular of the definition of [pred_strong1] may be surprising. We took advantage of [Definition]'s syntactic sugar for defining function arguments in the case of [n], but we bound the proofs later with explicit [fun] expressions. Let us see what happens if we write this function in the way that at first seems most natural.
adamc@70 89
adam@440 90 %\vspace{-.15in}%[[
adamc@70 91 Definition pred_strong1' (n : nat) (pf : n > 0) : nat :=
adamc@70 92 match n with
adamc@70 93 | O => match zgtz pf with end
adamc@70 94 | S n' => n'
adamc@70 95 end.
adam@335 96 ]]
adamc@70 97
adam@335 98 <<
adamc@70 99 Error: In environment
adamc@70 100 n : nat
adamc@70 101 pf : n > 0
adamc@70 102 The term "pf" has type "n > 0" while it is expected to have type
adamc@70 103 "0 > 0"
adam@335 104 >>
adamc@70 105
adamc@212 106 The term [zgtz pf] fails to type-check. Somehow the type checker has failed to take into account information that follows from which [match] branch that term appears in. The problem is that, by default, [match] does not let us use such implied information. To get refined typing, we must always rely on [match] annotations, either written explicitly or inferred.
adamc@70 107
adam@398 108 In this case, we must use a [return] annotation to declare the relationship between the _value_ of the [match] discriminee and the _type_ of the result. There is no annotation that lets us declare a relationship between the discriminee and the type of a variable that is already in scope; hence, we delay the binding of [pf], so that we can use the [return] annotation to express the needed relationship.
adamc@70 109
adam@471 110 We are lucky that Coq's heuristics infer the [return] clause (specifically, [return n > 0 -> nat]) for us in the definition of [pred_strong1], leading to the following elaborated code: *)
adam@335 111
adam@335 112 Definition pred_strong1' (n : nat) : n > 0 -> nat :=
adam@335 113 match n return n > 0 -> nat with
adam@335 114 | O => fun pf : 0 > 0 => match zgtz pf with end
adam@335 115 | S n' => fun _ => n'
adam@335 116 end.
adam@335 117
adam@403 118 (** By making explicit the functional relationship between value [n] and the result type of the [match], we guide Coq toward proper type checking. The clause for this example follows by simple copying of the original annotation on the definition. In general, however, the [match] annotation inference problem is undecidable. The known undecidable problem of%\index{higher-order unification}% _higher-order unification_ %\cite{HOU}% reduces to the [match] type inference problem. Over time, Coq is enhanced with more and more heuristics to get around this problem, but there must always exist [match]es whose types Coq cannot infer without annotations.
adamc@70 119
adamc@70 120 Let us now take a look at the OCaml code Coq generates for [pred_strong1]. *)
adamc@70 121
adamc@70 122 Extraction pred_strong1.
adamc@70 123
adamc@70 124 (** %\begin{verbatim}
adamc@70 125 (** val pred_strong1 : nat -> nat **)
adamc@70 126
adamc@70 127 let pred_strong1 = function
adamc@70 128 | O -> assert false (* absurd case *)
adamc@70 129 | S n' -> n'
adamc@70 130 \end{verbatim}%
adamc@70 131
adamc@70 132 #<pre>
adamc@70 133 (** val pred_strong1 : nat -> nat **)
adamc@70 134
adamc@70 135 let pred_strong1 = function
adamc@70 136 | O -> assert false (* absurd case *)
adamc@70 137 | S n' -> n'
adamc@70 138 </pre># *)
adamc@70 139
adam@451 140 (** The proof argument has disappeared! We get exactly the OCaml code we would have written manually. This is our first demonstration of the main technically interesting feature of Coq program extraction: proofs are erased systematically.
adamc@70 141
adam@471 142 %\medskip%
adam@471 143
adam@403 144 We can reimplement our dependently typed [pred] based on%\index{subset types}% _subset types_, defined in the standard library with the type family %\index{Gallina terms!sig}%[sig]. *)
adamc@70 145
adam@423 146 (* begin hide *)
adam@437 147 (* begin thide *)
adam@437 148 Definition bar := ex.
adam@437 149 (* end thide *)
adam@423 150 (* end hide *)
adam@423 151
adamc@70 152 Print sig.
adamc@212 153 (** %\vspace{-.15in}% [[
adamc@70 154 Inductive sig (A : Type) (P : A -> Prop) : Type :=
adamc@70 155 exist : forall x : A, P x -> sig P
adamc@70 156 ]]
adamc@70 157
adam@442 158 The family [sig] is a Curry-Howard twin of [ex], except that [sig] is in [Type], while [ex] is in [Prop]. That means that [sig] values can survive extraction, while [ex] proofs will always be erased. The actual details of extraction of [sig]s are more subtle, as we will see shortly.
adamc@70 159
adamc@70 160 We rewrite [pred_strong1], using some syntactic sugar for subset types. *)
adamc@70 161
adamc@70 162 Locate "{ _ : _ | _ }".
adamc@212 163 (** %\vspace{-.15in}% [[
adam@495 164 Notation
adam@495 165 "{ x : A | P }" := sig (fun x : A => P)
adam@495 166 ]]
adam@302 167 *)
adamc@70 168
adamc@70 169 Definition pred_strong2 (s : {n : nat | n > 0}) : nat :=
adamc@70 170 match s with
adamc@70 171 | exist O pf => match zgtz pf with end
adamc@70 172 | exist (S n') _ => n'
adamc@70 173 end.
adamc@70 174
adam@474 175 (** To build a value of a subset type, we use the [exist] constructor, and the details of how to do that follow from the output of our earlier [Print sig] command, where we elided the extra information that parameter [A] is implicit. We need an extra [_] here and not in the definition of [pred_strong2] because _parameters_ of inductive types (like the predicate [P] for [sig]) are not mentioned in pattern matching, but _are_ mentioned in construction of terms (if they are not marked as implicit arguments). *)
adam@282 176
adam@282 177 Eval compute in pred_strong2 (exist _ 2 two_gt0).
adam@282 178 (** %\vspace{-.15in}% [[
adam@282 179 = 1
adam@282 180 : nat
adam@302 181 ]]
adam@302 182 *)
adam@282 183
adamc@70 184 Extraction pred_strong2.
adamc@70 185
adamc@70 186 (** %\begin{verbatim}
adamc@70 187 (** val pred_strong2 : nat -> nat **)
adamc@70 188
adamc@70 189 let pred_strong2 = function
adamc@70 190 | O -> assert false (* absurd case *)
adamc@70 191 | S n' -> n'
adamc@70 192 \end{verbatim}%
adamc@70 193
adamc@70 194 #<pre>
adamc@70 195 (** val pred_strong2 : nat -> nat **)
adamc@70 196
adamc@70 197 let pred_strong2 = function
adamc@70 198 | O -> assert false (* absurd case *)
adamc@70 199 | S n' -> n'
adamc@70 200 </pre>#
adamc@70 201
adamc@70 202 We arrive at the same OCaml code as was extracted from [pred_strong1], which may seem surprising at first. The reason is that a value of [sig] is a pair of two pieces, a value and a proof about it. Extraction erases the proof, which reduces the constructor [exist] of [sig] to taking just a single argument. An optimization eliminates uses of datatypes with single constructors taking single arguments, and we arrive back where we started.
adamc@70 203
adamc@70 204 We can continue on in the process of refining [pred]'s type. Let us change its result type to capture that the output is really the predecessor of the input. *)
adamc@70 205
adamc@70 206 Definition pred_strong3 (s : {n : nat | n > 0}) : {m : nat | proj1_sig s = S m} :=
adamc@70 207 match s return {m : nat | proj1_sig s = S m} with
adamc@70 208 | exist 0 pf => match zgtz pf with end
adam@426 209 | exist (S n') pf => exist _ n' (eq_refl _)
adamc@70 210 end.
adamc@70 211
adam@495 212 (* begin hide *)
adam@495 213 (* begin thide *)
adam@495 214 Definition ugh := lt.
adam@495 215 (* end thide *)
adam@495 216 (* end hide *)
adam@495 217
adam@282 218 Eval compute in pred_strong3 (exist _ 2 two_gt0).
adam@282 219 (** %\vspace{-.15in}% [[
adam@426 220 = exist (fun m : nat => 2 = S m) 1 (eq_refl 2)
adam@282 221 : {m : nat | proj1_sig (exist (lt 0) 2 two_gt0) = S m}
adam@335 222 ]]
adam@302 223 *)
adam@282 224
adam@423 225 (* begin hide *)
adam@437 226 (* begin thide *)
adam@423 227 Definition pred_strong := 0.
adam@437 228 (* end thide *)
adam@423 229 (* end hide *)
adam@423 230
adam@474 231 (** A value in a subset type can be thought of as a%\index{dependent pair}% _dependent pair_ (or%\index{sigma type}% _sigma type_) of a base value and a proof about it. The function %\index{Gallina terms!proj1\_sig}%[proj1_sig] extracts the first component of the pair. It turns out that we need to include an explicit [return] clause here, since Coq's heuristics are not smart enough to propagate the result type that we wrote earlier.
adamc@70 232
adamc@70 233 By now, the reader is probably ready to believe that the new [pred_strong] leads to the same OCaml code as we have seen several times so far, and Coq does not disappoint. *)
adamc@70 234
adamc@70 235 Extraction pred_strong3.
adamc@70 236
adamc@70 237 (** %\begin{verbatim}
adamc@70 238 (** val pred_strong3 : nat -> nat **)
adamc@70 239
adamc@70 240 let pred_strong3 = function
adamc@70 241 | O -> assert false (* absurd case *)
adamc@70 242 | S n' -> n'
adamc@70 243 \end{verbatim}%
adamc@70 244
adamc@70 245 #<pre>
adamc@70 246 (** val pred_strong3 : nat -> nat **)
adamc@70 247
adamc@70 248 let pred_strong3 = function
adamc@70 249 | O -> assert false (* absurd case *)
adamc@70 250 | S n' -> n'
adamc@70 251 </pre>#
adamc@70 252
adam@335 253 We have managed to reach a type that is, in a formal sense, the most expressive possible for [pred]. Any other implementation of the same type must have the same input-output behavior. However, there is still room for improvement in making this kind of code easier to write. Here is a version that takes advantage of tactic-based theorem proving. We switch back to passing a separate proof argument instead of using a subset type for the function's input, because this leads to cleaner code. (Recall that [False_rec] is the [Set]-level induction principle for [False], which can be used to produce a value in any [Set] given a proof of [False].) *)
adamc@70 254
adam@297 255 Definition pred_strong4 : forall n : nat, n > 0 -> {m : nat | n = S m}.
adamc@70 256 refine (fun n =>
adamc@212 257 match n with
adamc@70 258 | O => fun _ => False_rec _ _
adamc@70 259 | S n' => fun _ => exist _ n' _
adamc@70 260 end).
adamc@212 261
adamc@77 262 (* begin thide *)
adam@335 263 (** We build [pred_strong4] using tactic-based proving, beginning with a [Definition] command that ends in a period before a definition is given. Such a command enters the interactive proving mode, with the type given for the new identifier as our proof goal. It may seem strange to change perspective so implicitly between programming and proving, but recall that programs and proofs are two sides of the same coin in Coq, thanks to the Curry-Howard correspondence.
adamc@70 264
adam@423 265 We do most of the work with the %\index{tactics!refine}%[refine] tactic, to which we pass a partial "proof" of the type we are trying to prove. There may be some pieces left to fill in, indicated by underscores. Any underscore that Coq cannot reconstruct with type inference is added as a proof subgoal. In this case, we have two subgoals:
adam@335 266
adam@335 267 [[
adam@439 268 2 subgoals
adamc@70 269
adamc@70 270 n : nat
adamc@70 271 _ : 0 > 0
adamc@70 272 ============================
adamc@70 273 False
adam@439 274
adam@439 275 subgoal 2 is
adam@439 276
adamc@70 277 S n' = S n'
adamc@70 278 ]]
adamc@70 279
adamc@70 280 We can see that the first subgoal comes from the second underscore passed to [False_rec], and the second subgoal comes from the second underscore passed to [exist]. In the first case, we see that, though we bound the proof variable with an underscore, it is still available in our proof context. It is hard to refer to underscore-named variables in manual proofs, but automation makes short work of them. Both subgoals are easy to discharge that way, so let us back up and ask to prove all subgoals automatically. *)
adamc@70 281
adamc@70 282 Undo.
adamc@70 283 refine (fun n =>
adamc@212 284 match n with
adamc@70 285 | O => fun _ => False_rec _ _
adamc@70 286 | S n' => fun _ => exist _ n' _
adamc@70 287 end); crush.
adamc@77 288 (* end thide *)
adamc@70 289 Defined.
adamc@70 290
adam@423 291 (** We end the "proof" with %\index{Vernacular commands!Defined}%[Defined] instead of [Qed], so that the definition we constructed remains visible. This contrasts to the case of ending a proof with [Qed], where the details of the proof are hidden afterward. (More formally, [Defined] marks an identifier as%\index{transparent}% _transparent_, allowing it to be unfolded; while [Qed] marks an identifier as%\index{opaque}% _opaque_, preventing unfolding.) Let us see what our proof script constructed. *)
adamc@70 292
adamc@70 293 Print pred_strong4.
adamc@212 294 (** %\vspace{-.15in}% [[
adamc@70 295 pred_strong4 =
adamc@70 296 fun n : nat =>
adamc@70 297 match n as n0 return (n0 > 0 -> {m : nat | n0 = S m}) with
adamc@70 298 | 0 =>
adamc@70 299 fun _ : 0 > 0 =>
adamc@70 300 False_rec {m : nat | 0 = S m}
adamc@70 301 (Bool.diff_false_true
adamc@70 302 (Bool.absurd_eq_true false
adamc@70 303 (Bool.diff_false_true
adamc@70 304 (Bool.absurd_eq_true false (pred_strong4_subproof n _)))))
adamc@70 305 | S n' =>
adamc@70 306 fun _ : S n' > 0 =>
adam@426 307 exist (fun m : nat => S n' = S m) n' (eq_refl (S n'))
adamc@70 308 end
adamc@70 309 : forall n : nat, n > 0 -> {m : nat | n = S m}
adamc@70 310 ]]
adamc@70 311
adam@442 312 We see the code we entered, with some proofs filled in. The first proof obligation, the second argument to [False_rec], is filled in with a nasty-looking proof term that we can be glad we did not enter by hand. The second proof obligation is a simple reflexivity proof. *)
adamc@70 313
adam@282 314 Eval compute in pred_strong4 two_gt0.
adam@282 315 (** %\vspace{-.15in}% [[
adam@426 316 = exist (fun m : nat => 2 = S m) 1 (eq_refl 2)
adam@282 317 : {m : nat | 2 = S m}
adam@282 318 ]]
adam@282 319
adam@442 320 A tactic modifier called %\index{tactics!abstract}%[abstract] can be helpful for producing shorter terms, by automatically abstracting subgoals into named lemmas. *)
adam@335 321
adam@335 322 (* begin thide *)
adam@335 323 Definition pred_strong4' : forall n : nat, n > 0 -> {m : nat | n = S m}.
adam@335 324 refine (fun n =>
adam@335 325 match n with
adam@335 326 | O => fun _ => False_rec _ _
adam@335 327 | S n' => fun _ => exist _ n' _
adam@335 328 end); abstract crush.
adam@335 329 Defined.
adam@335 330
adam@335 331 Print pred_strong4'.
adam@335 332 (* end thide *)
adam@335 333
adam@335 334 (** %\vspace{-.15in}% [[
adam@335 335 pred_strong4' =
adam@335 336 fun n : nat =>
adam@335 337 match n as n0 return (n0 > 0 -> {m : nat | n0 = S m}) with
adam@335 338 | 0 =>
adam@335 339 fun _H : 0 > 0 =>
adam@335 340 False_rec {m : nat | 0 = S m} (pred_strong4'_subproof n _H)
adam@335 341 | S n' =>
adam@335 342 fun _H : S n' > 0 =>
adam@335 343 exist (fun m : nat => S n' = S m) n' (pred_strong4'_subproof0 n _H)
adam@335 344 end
adam@335 345 : forall n : nat, n > 0 -> {m : nat | n = S m}
adam@335 346 ]]
adam@335 347
adam@338 348 We are almost done with the ideal implementation of dependent predecessor. We can use Coq's syntax extension facility to arrive at code with almost no complexity beyond a Haskell or ML program with a complete specification in a comment. In this book, we will not dwell on the details of syntax extensions; the Coq manual gives a straightforward introduction to them. *)
adamc@70 349
adamc@70 350 Notation "!" := (False_rec _ _).
adamc@70 351 Notation "[ e ]" := (exist _ e _).
adamc@70 352
adam@297 353 Definition pred_strong5 : forall n : nat, n > 0 -> {m : nat | n = S m}.
adamc@70 354 refine (fun n =>
adamc@212 355 match n with
adamc@70 356 | O => fun _ => !
adamc@70 357 | S n' => fun _ => [n']
adamc@70 358 end); crush.
adamc@70 359 Defined.
adamc@71 360
adam@282 361 (** By default, notations are also used in pretty-printing terms, including results of evaluation. *)
adam@282 362
adam@282 363 Eval compute in pred_strong5 two_gt0.
adam@282 364 (** %\vspace{-.15in}% [[
adam@282 365 = [1]
adam@282 366 : {m : nat | 2 = S m}
adam@282 367 ]]
adam@282 368
adam@442 369 One other alternative is worth demonstrating. Recent Coq versions include a facility called %\index{Program}%[Program] that streamlines this style of definition. Here is a complete implementation using [Program].%\index{Vernacular commands!Obligation Tactic}\index{Vernacular commands!Program Definition}% *)
adamc@212 370
adamc@212 371 Obligation Tactic := crush.
adamc@212 372
adamc@212 373 Program Definition pred_strong6 (n : nat) (_ : n > 0) : {m : nat | n = S m} :=
adamc@212 374 match n with
adamc@212 375 | O => _
adamc@212 376 | S n' => n'
adamc@212 377 end.
adamc@212 378
adam@495 379 (** Printing the resulting definition of [pred_strong6] yields a term very similar to what we built with [refine]. [Program] can save time in writing programs that use subset types. Nonetheless, [refine] is often just as effective, and [refine] gives more control over the form the final term takes, which can be useful when you want to prove additional theorems about your definition. [Program] will sometimes insert type casts that can complicate theorem proving. *)
adamc@212 380
adam@282 381 Eval compute in pred_strong6 two_gt0.
adam@282 382 (** %\vspace{-.15in}% [[
adam@282 383 = [1]
adam@282 384 : {m : nat | 2 = S m}
adam@302 385 ]]
adam@335 386
adam@442 387 In this case, we see that the new definition yields the same computational behavior as before. *)
adam@282 388
adamc@71 389
adamc@71 390 (** * Decidable Proposition Types *)
adamc@71 391
adam@495 392 (** There is another type in the standard library that captures the idea of program values that indicate which of two propositions is true.%\index{Gallina terms!sumbool}% *)
adamc@71 393
adamc@71 394 Print sumbool.
adamc@212 395 (** %\vspace{-.15in}% [[
adamc@71 396 Inductive sumbool (A : Prop) (B : Prop) : Set :=
adamc@71 397 left : A -> {A} + {B} | right : B -> {A} + {B}
adamc@212 398 ]]
adamc@71 399
adam@471 400 Here, the constructors of [sumbool] have types written in terms of a registered notation for [sumbool], such that the result type of each constructor desugars to [sumbool A B]. We can define some notations of our own to make working with [sumbool] more convenient. *)
adamc@71 401
adamc@71 402 Notation "'Yes'" := (left _ _).
adamc@71 403 Notation "'No'" := (right _ _).
adamc@71 404 Notation "'Reduce' x" := (if x then Yes else No) (at level 50).
adamc@71 405
adam@436 406 (** The %\coqdocnotation{%#<tt>#Reduce#</tt>#%}% notation is notable because it demonstrates how [if] is overloaded in Coq. The [if] form actually works when the test expression has any two-constructor inductive type. Moreover, in the [then] and [else] branches, the appropriate constructor arguments are bound. This is important when working with [sumbool]s, when we want to have the proof stored in the test expression available when proving the proof obligations generated in the appropriate branch.
adamc@71 407
adamc@71 408 Now we can write [eq_nat_dec], which compares two natural numbers, returning either a proof of their equality or a proof of their inequality. *)
adamc@71 409
adam@297 410 Definition eq_nat_dec : forall n m : nat, {n = m} + {n <> m}.
adamc@212 411 refine (fix f (n m : nat) : {n = m} + {n <> m} :=
adamc@212 412 match n, m with
adamc@71 413 | O, O => Yes
adamc@71 414 | S n', S m' => Reduce (f n' m')
adamc@71 415 | _, _ => No
adamc@71 416 end); congruence.
adamc@71 417 Defined.
adamc@71 418
adam@282 419 Eval compute in eq_nat_dec 2 2.
adam@282 420 (** %\vspace{-.15in}% [[
adam@282 421 = Yes
adam@282 422 : {2 = 2} + {2 <> 2}
adam@302 423 ]]
adam@302 424 *)
adam@282 425
adam@282 426 Eval compute in eq_nat_dec 2 3.
adam@282 427 (** %\vspace{-.15in}% [[
adam@282 428 = No
adam@341 429 : {2 = 3} + {2 <> 3}
adam@302 430 ]]
adam@282 431
adam@442 432 Note that the %\coqdocnotation{%#<tt>#Yes#</tt>#%}% and %\coqdocnotation{%#<tt>#No#</tt>#%}% notations are hiding proofs establishing the correctness of the outputs.
adam@335 433
adam@335 434 Our definition extracts to reasonable OCaml code. *)
adamc@71 435
adamc@71 436 Extraction eq_nat_dec.
adamc@71 437
adamc@71 438 (** %\begin{verbatim}
adamc@71 439 (** val eq_nat_dec : nat -> nat -> sumbool **)
adamc@71 440
adamc@71 441 let rec eq_nat_dec n m =
adamc@71 442 match n with
adamc@71 443 | O -> (match m with
adamc@71 444 | O -> Left
adamc@71 445 | S n0 -> Right)
adamc@71 446 | S n' -> (match m with
adamc@71 447 | O -> Right
adamc@71 448 | S m' -> eq_nat_dec n' m')
adamc@71 449 \end{verbatim}%
adamc@71 450
adamc@71 451 #<pre>
adamc@71 452 (** val eq_nat_dec : nat -> nat -> sumbool **)
adamc@71 453
adamc@71 454 let rec eq_nat_dec n m =
adamc@71 455 match n with
adamc@71 456 | O -> (match m with
adamc@71 457 | O -> Left
adamc@71 458 | S n0 -> Right)
adamc@71 459 | S n' -> (match m with
adamc@71 460 | O -> Right
adamc@71 461 | S m' -> eq_nat_dec n' m')
adamc@71 462 </pre>#
adamc@71 463
adam@335 464 Proving this kind of decidable equality result is so common that Coq comes with a tactic for automating it.%\index{tactics!decide equality}% *)
adamc@71 465
adamc@71 466 Definition eq_nat_dec' (n m : nat) : {n = m} + {n <> m}.
adamc@71 467 decide equality.
adamc@71 468 Defined.
adamc@71 469
adam@448 470 (** Curious readers can verify that the [decide equality] version extracts to the same OCaml code as our more manual version does. That OCaml code had one undesirable property, which is that it uses <<Left>> and <<Right>> constructors instead of the Boolean values built into OCaml. We can fix this, by using Coq's facility for mapping Coq inductive types to OCaml variant types.%\index{Vernacular commands!Extract Inductive}% *)
adamc@71 471
adamc@71 472 Extract Inductive sumbool => "bool" ["true" "false"].
adamc@71 473 Extraction eq_nat_dec'.
adamc@71 474
adamc@71 475 (** %\begin{verbatim}
adamc@71 476 (** val eq_nat_dec' : nat -> nat -> bool **)
adamc@71 477
adamc@71 478 let rec eq_nat_dec' n m0 =
adamc@71 479 match n with
adamc@71 480 | O -> (match m0 with
adamc@71 481 | O -> true
adamc@71 482 | S n0 -> false)
adamc@71 483 | S n0 -> (match m0 with
adamc@71 484 | O -> false
adamc@71 485 | S n1 -> eq_nat_dec' n0 n1)
adamc@71 486 \end{verbatim}%
adamc@71 487
adamc@71 488 #<pre>
adamc@71 489 (** val eq_nat_dec' : nat -> nat -> bool **)
adamc@71 490
adamc@71 491 let rec eq_nat_dec' n m0 =
adamc@71 492 match n with
adamc@71 493 | O -> (match m0 with
adamc@71 494 | O -> true
adamc@71 495 | S n0 -> false)
adamc@71 496 | S n0 -> (match m0 with
adamc@71 497 | O -> false
adamc@71 498 | S n1 -> eq_nat_dec' n0 n1)
adamc@71 499 </pre># *)
adamc@72 500
adamc@72 501 (** %\smallskip%
adamc@72 502
adam@448 503 We can build "smart" versions of the usual Boolean operators and put them to good use in certified programming. For instance, here is a [sumbool] version of Boolean "or." *)
adamc@72 504
adam@337 505 (* EX: Write a function that decides if an element belongs to a list. *)
adam@337 506
adamc@77 507 (* begin thide *)
adamc@204 508 Notation "x || y" := (if x then Yes else Reduce y).
adamc@72 509
adamc@72 510 (** Let us use it for building a function that decides list membership. We need to assume the existence of an equality decision procedure for the type of list elements. *)
adamc@72 511
adamc@72 512 Section In_dec.
adamc@72 513 Variable A : Set.
adamc@72 514 Variable A_eq_dec : forall x y : A, {x = y} + {x <> y}.
adamc@72 515
adamc@72 516 (** The final function is easy to write using the techniques we have developed so far. *)
adamc@72 517
adamc@212 518 Definition In_dec : forall (x : A) (ls : list A), {In x ls} + {~ In x ls}.
adamc@212 519 refine (fix f (x : A) (ls : list A) : {In x ls} + {~ In x ls} :=
adamc@212 520 match ls with
adamc@72 521 | nil => No
adamc@72 522 | x' :: ls' => A_eq_dec x x' || f x ls'
adamc@72 523 end); crush.
adam@282 524 Defined.
adamc@72 525 End In_dec.
adamc@72 526
adam@282 527 Eval compute in In_dec eq_nat_dec 2 (1 :: 2 :: nil).
adam@282 528 (** %\vspace{-.15in}% [[
adam@282 529 = Yes
adam@469 530 : {In 2 (1 :: 2 :: nil)} + { ~ In 2 (1 :: 2 :: nil)}
adam@302 531 ]]
adam@302 532 *)
adam@282 533
adam@282 534 Eval compute in In_dec eq_nat_dec 3 (1 :: 2 :: nil).
adam@282 535 (** %\vspace{-.15in}% [[
adam@282 536 = No
adam@469 537 : {In 3 (1 :: 2 :: nil)} + { ~ In 3 (1 :: 2 :: nil)}
adam@302 538 ]]
adam@282 539
adam@469 540 The [In_dec] function has a reasonable extraction to OCaml. *)
adamc@72 541
adamc@72 542 Extraction In_dec.
adamc@77 543 (* end thide *)
adamc@72 544
adamc@72 545 (** %\begin{verbatim}
adamc@72 546 (** val in_dec : ('a1 -> 'a1 -> bool) -> 'a1 -> 'a1 list -> bool **)
adamc@72 547
adamc@72 548 let rec in_dec a_eq_dec x = function
adamc@72 549 | Nil -> false
adamc@72 550 | Cons (x', ls') ->
adamc@72 551 (match a_eq_dec x x' with
adamc@72 552 | true -> true
adamc@72 553 | false -> in_dec a_eq_dec x ls')
adamc@72 554 \end{verbatim}%
adamc@72 555
adamc@72 556 #<pre>
adamc@72 557 (** val in_dec : ('a1 -> 'a1 -> bool) -> 'a1 -> 'a1 list -> bool **)
adamc@72 558
adamc@72 559 let rec in_dec a_eq_dec x = function
adamc@72 560 | Nil -> false
adamc@72 561 | Cons (x', ls') ->
adamc@72 562 (match a_eq_dec x x' with
adamc@72 563 | true -> true
adamc@72 564 | false -> in_dec a_eq_dec x ls')
adam@403 565 </pre>#
adam@403 566
adam@403 567 This is more or the less code for the corresponding function from the OCaml standard library. *)
adamc@72 568
adamc@72 569
adamc@72 570 (** * Partial Subset Types *)
adamc@72 571
adam@335 572 (** Our final implementation of dependent predecessor used a very specific argument type to ensure that execution could always complete normally. Sometimes we want to allow execution to fail, and we want a more principled way of signaling failure than returning a default value, as [pred] does for [0]. One approach is to define this type family %\index{Gallina terms!maybe}%[maybe], which is a version of [sig] that allows obligation-free failure. *)
adamc@73 573
adamc@89 574 Inductive maybe (A : Set) (P : A -> Prop) : Set :=
adamc@72 575 | Unknown : maybe P
adamc@72 576 | Found : forall x : A, P x -> maybe P.
adamc@72 577
adamc@73 578 (** We can define some new notations, analogous to those we defined for subset types. *)
adamc@73 579
adamc@72 580 Notation "{{ x | P }}" := (maybe (fun x => P)).
adamc@72 581 Notation "??" := (Unknown _).
adam@335 582 Notation "[| x |]" := (Found _ x _).
adamc@72 583
adamc@73 584 (** Now our next version of [pred] is trivial to write. *)
adamc@73 585
adam@297 586 Definition pred_strong7 : forall n : nat, {{m | n = S m}}.
adamc@73 587 refine (fun n =>
adam@380 588 match n return {{m | n = S m}} with
adamc@73 589 | O => ??
adam@335 590 | S n' => [|n'|]
adamc@73 591 end); trivial.
adamc@73 592 Defined.
adamc@73 593
adam@282 594 Eval compute in pred_strong7 2.
adam@282 595 (** %\vspace{-.15in}% [[
adam@335 596 = [|1|]
adam@282 597 : {{m | 2 = S m}}
adam@335 598 ]]
adam@302 599 *)
adam@282 600
adam@282 601 Eval compute in pred_strong7 0.
adam@282 602 (** %\vspace{-.15in}% [[
adam@282 603 = ??
adam@282 604 : {{m | 0 = S m}}
adam@282 605 ]]
adam@282 606
adam@442 607 Because we used [maybe], one valid implementation of the type we gave [pred_strong7] would return [??] in every case. We can strengthen the type to rule out such vacuous implementations, and the type family %\index{Gallina terms!sumor}%[sumor] from the standard library provides the easiest starting point. For type [A] and proposition [B], [A + {B}] desugars to [sumor A B], whose values are either values of [A] or proofs of [B]. *)
adamc@73 608
adamc@73 609 Print sumor.
adamc@212 610 (** %\vspace{-.15in}% [[
adamc@73 611 Inductive sumor (A : Type) (B : Prop) : Type :=
adamc@73 612 inleft : A -> A + {B} | inright : B -> A + {B}
adam@302 613 ]]
adamc@73 614
adam@442 615 We add notations for easy use of the [sumor] constructors. The second notation is specialized to [sumor]s whose [A] parameters are instantiated with regular subset types, since this is how we will use [sumor] below. *)
adamc@73 616
adamc@73 617 Notation "!!" := (inright _ _).
adam@335 618 Notation "[|| x ||]" := (inleft _ [x]).
adamc@73 619
adam@335 620 (** Now we are ready to give the final version of possibly failing predecessor. The [sumor]-based type that we use is maximally expressive; any implementation of the type has the same input-output behavior. *)
adamc@73 621
adam@297 622 Definition pred_strong8 : forall n : nat, {m : nat | n = S m} + {n = 0}.
adamc@73 623 refine (fun n =>
adamc@212 624 match n with
adamc@73 625 | O => !!
adam@335 626 | S n' => [||n'||]
adamc@73 627 end); trivial.
adamc@73 628 Defined.
adamc@73 629
adam@282 630 Eval compute in pred_strong8 2.
adam@282 631 (** %\vspace{-.15in}% [[
adam@335 632 = [||1||]
adam@282 633 : {m : nat | 2 = S m} + {2 = 0}
adam@302 634 ]]
adam@302 635 *)
adam@282 636
adam@282 637 Eval compute in pred_strong8 0.
adam@282 638 (** %\vspace{-.15in}% [[
adam@282 639 = !!
adam@282 640 : {m : nat | 0 = S m} + {0 = 0}
adam@302 641 ]]
adam@302 642 *)
adam@282 643
adam@335 644 (** As with our other maximally expressive [pred] function, we arrive at quite simple output values, thanks to notations. *)
adam@335 645
adamc@73 646
adamc@73 647 (** * Monadic Notations *)
adamc@73 648
adam@471 649 (** We can treat [maybe] like a monad%~\cite{Monads}\index{monad}\index{failure monad}%, in the same way that the Haskell <<Maybe>> type is interpreted as a failure monad. Our [maybe] has the wrong type to be a literal monad, but a "bind"-like notation will still be helpful. %Note that the notation definition uses an ASCII \texttt{<-}, while later code uses (in this rendering) a nicer left arrow $\leftarrow$.% *)
adamc@73 650
adamc@72 651 Notation "x <- e1 ; e2" := (match e1 with
adamc@72 652 | Unknown => ??
adamc@72 653 | Found x _ => e2
adamc@72 654 end)
adamc@72 655 (right associativity, at level 60).
adamc@72 656
adam@398 657 (** The meaning of [x <- e1; e2] is: First run [e1]. If it fails to find an answer, then announce failure for our derived computation, too. If [e1] _does_ find an answer, pass that answer on to [e2] to find the final result. The variable [x] can be considered bound in [e2].
adamc@73 658
adam@335 659 This notation is very helpful for composing richly typed procedures. For instance, here is a very simple implementation of a function to take the predecessors of two naturals at once. *)
adamc@73 660
adam@337 661 (* EX: Write a function that tries to compute predecessors of two [nat]s at once. *)
adam@337 662
adam@337 663 (* begin thide *)
adam@297 664 Definition doublePred : forall n1 n2 : nat, {{p | n1 = S (fst p) /\ n2 = S (snd p)}}.
adamc@73 665 refine (fun n1 n2 =>
adamc@212 666 m1 <- pred_strong7 n1;
adamc@212 667 m2 <- pred_strong7 n2;
adam@335 668 [|(m1, m2)|]); tauto.
adamc@73 669 Defined.
adam@337 670 (* end thide *)
adamc@73 671
adam@471 672 (** We can build a [sumor] version of the "bind" notation and use it to write a similarly straightforward version of this function. %Again, the notation definition exposes the ASCII syntax with an operator \texttt{<-{}-}, while the later code uses a nicer long left arrow $\longleftarrow$.% *)
adamc@73 673
adamc@73 674 Notation "x <-- e1 ; e2" := (match e1 with
adamc@73 675 | inright _ => !!
adamc@73 676 | inleft (exist x _) => e2
adamc@73 677 end)
adamc@73 678 (right associativity, at level 60).
adamc@73 679
adamc@73 680 (** printing * $\times$ *)
adamc@73 681
adam@337 682 (* EX: Write a more expressively typed version of the last exercise. *)
adam@337 683
adam@337 684 (* begin thide *)
adam@297 685 Definition doublePred' : forall n1 n2 : nat,
adam@297 686 {p : nat * nat | n1 = S (fst p) /\ n2 = S (snd p)}
adamc@73 687 + {n1 = 0 \/ n2 = 0}.
adamc@73 688 refine (fun n1 n2 =>
adamc@212 689 m1 <-- pred_strong8 n1;
adamc@212 690 m2 <-- pred_strong8 n2;
adam@335 691 [||(m1, m2)||]); tauto.
adamc@73 692 Defined.
adam@337 693 (* end thide *)
adamc@72 694
adam@392 695 (** This example demonstrates how judicious selection of notations can hide complexities in the rich types of programs. *)
adam@392 696
adamc@72 697
adamc@72 698 (** * A Type-Checking Example *)
adamc@72 699
adam@335 700 (** We can apply these specification types to build a certified type checker for a simple expression language. *)
adamc@75 701
adamc@72 702 Inductive exp : Set :=
adamc@72 703 | Nat : nat -> exp
adamc@72 704 | Plus : exp -> exp -> exp
adamc@72 705 | Bool : bool -> exp
adamc@72 706 | And : exp -> exp -> exp.
adamc@72 707
adamc@75 708 (** We define a simple language of types and its typing rules, in the style introduced in Chapter 4. *)
adamc@75 709
adamc@72 710 Inductive type : Set := TNat | TBool.
adamc@72 711
adamc@72 712 Inductive hasType : exp -> type -> Prop :=
adamc@72 713 | HtNat : forall n,
adamc@72 714 hasType (Nat n) TNat
adamc@72 715 | HtPlus : forall e1 e2,
adamc@72 716 hasType e1 TNat
adamc@72 717 -> hasType e2 TNat
adamc@72 718 -> hasType (Plus e1 e2) TNat
adamc@72 719 | HtBool : forall b,
adamc@72 720 hasType (Bool b) TBool
adamc@72 721 | HtAnd : forall e1 e2,
adamc@72 722 hasType e1 TBool
adamc@72 723 -> hasType e2 TBool
adamc@72 724 -> hasType (And e1 e2) TBool.
adamc@72 725
adamc@75 726 (** It will be helpful to have a function for comparing two types. We build one using [decide equality]. *)
adamc@75 727
adamc@77 728 (* begin thide *)
adamc@75 729 Definition eq_type_dec : forall t1 t2 : type, {t1 = t2} + {t1 <> t2}.
adamc@72 730 decide equality.
adamc@72 731 Defined.
adamc@72 732
adam@423 733 (** Another notation complements the monadic notation for [maybe] that we defined earlier. Sometimes we want to include "assertions" in our procedures. That is, we want to run a decision procedure and fail if it fails; otherwise, we want to continue, with the proof that it produced made available to us. This infix notation captures that idea, for a procedure that returns an arbitrary two-constructor type. *)
adamc@75 734
adamc@73 735 Notation "e1 ;; e2" := (if e1 then e2 else ??)
adamc@73 736 (right associativity, at level 60).
adamc@73 737
adam@335 738 (** With that notation defined, we can implement a [typeCheck] function, whose code is only more complex than what we would write in ML because it needs to include some extra type annotations. Every [[|e|]] expression adds a [hasType] proof obligation, and [crush] makes short work of them when we add [hasType]'s constructors as hints. *)
adamc@77 739 (* end thide *)
adamc@75 740
adam@297 741 Definition typeCheck : forall e : exp, {{t | hasType e t}}.
adamc@77 742 (* begin thide *)
adamc@72 743 Hint Constructors hasType.
adamc@72 744
adamc@72 745 refine (fix F (e : exp) : {{t | hasType e t}} :=
adam@380 746 match e return {{t | hasType e t}} with
adam@335 747 | Nat _ => [|TNat|]
adamc@72 748 | Plus e1 e2 =>
adamc@72 749 t1 <- F e1;
adamc@72 750 t2 <- F e2;
adamc@72 751 eq_type_dec t1 TNat;;
adamc@72 752 eq_type_dec t2 TNat;;
adam@335 753 [|TNat|]
adam@335 754 | Bool _ => [|TBool|]
adamc@72 755 | And e1 e2 =>
adamc@72 756 t1 <- F e1;
adamc@72 757 t2 <- F e2;
adamc@72 758 eq_type_dec t1 TBool;;
adamc@72 759 eq_type_dec t2 TBool;;
adam@335 760 [|TBool|]
adamc@72 761 end); crush.
adamc@77 762 (* end thide *)
adamc@72 763 Defined.
adamc@72 764
adamc@75 765 (** Despite manipulating proofs, our type checker is easy to run. *)
adamc@75 766
adamc@72 767 Eval simpl in typeCheck (Nat 0).
adamc@212 768 (** %\vspace{-.15in}% [[
adam@335 769 = [|TNat|]
adamc@75 770 : {{t | hasType (Nat 0) t}}
adam@302 771 ]]
adam@302 772 *)
adamc@75 773
adamc@72 774 Eval simpl in typeCheck (Plus (Nat 1) (Nat 2)).
adamc@212 775 (** %\vspace{-.15in}% [[
adam@335 776 = [|TNat|]
adamc@75 777 : {{t | hasType (Plus (Nat 1) (Nat 2)) t}}
adam@302 778 ]]
adam@302 779 *)
adamc@75 780
adamc@72 781 Eval simpl in typeCheck (Plus (Nat 1) (Bool false)).
adamc@212 782 (** %\vspace{-.15in}% [[
adamc@75 783 = ??
adamc@75 784 : {{t | hasType (Plus (Nat 1) (Bool false)) t}}
adam@302 785 ]]
adamc@75 786
adam@442 787 The type checker also extracts to some reasonable OCaml code. *)
adamc@75 788
adamc@75 789 Extraction typeCheck.
adamc@75 790
adamc@75 791 (** %\begin{verbatim}
adamc@75 792 (** val typeCheck : exp -> type0 maybe **)
adamc@75 793
adamc@75 794 let rec typeCheck = function
adamc@75 795 | Nat n -> Found TNat
adamc@75 796 | Plus (e1, e2) ->
adamc@75 797 (match typeCheck e1 with
adamc@75 798 | Unknown -> Unknown
adamc@75 799 | Found t1 ->
adamc@75 800 (match typeCheck e2 with
adamc@75 801 | Unknown -> Unknown
adamc@75 802 | Found t2 ->
adamc@75 803 (match eq_type_dec t1 TNat with
adamc@75 804 | true ->
adamc@75 805 (match eq_type_dec t2 TNat with
adamc@75 806 | true -> Found TNat
adamc@75 807 | false -> Unknown)
adamc@75 808 | false -> Unknown)))
adamc@75 809 | Bool b -> Found TBool
adamc@75 810 | And (e1, e2) ->
adamc@75 811 (match typeCheck e1 with
adamc@75 812 | Unknown -> Unknown
adamc@75 813 | Found t1 ->
adamc@75 814 (match typeCheck e2 with
adamc@75 815 | Unknown -> Unknown
adamc@75 816 | Found t2 ->
adamc@75 817 (match eq_type_dec t1 TBool with
adamc@75 818 | true ->
adamc@75 819 (match eq_type_dec t2 TBool with
adamc@75 820 | true -> Found TBool
adamc@75 821 | false -> Unknown)
adamc@75 822 | false -> Unknown)))
adamc@75 823 \end{verbatim}%
adamc@75 824
adamc@75 825 #<pre>
adamc@75 826 (** val typeCheck : exp -> type0 maybe **)
adamc@75 827
adamc@75 828 let rec typeCheck = function
adamc@75 829 | Nat n -> Found TNat
adamc@75 830 | Plus (e1, e2) ->
adamc@75 831 (match typeCheck e1 with
adamc@75 832 | Unknown -> Unknown
adamc@75 833 | Found t1 ->
adamc@75 834 (match typeCheck e2 with
adamc@75 835 | Unknown -> Unknown
adamc@75 836 | Found t2 ->
adamc@75 837 (match eq_type_dec t1 TNat with
adamc@75 838 | true ->
adamc@75 839 (match eq_type_dec t2 TNat with
adamc@75 840 | true -> Found TNat
adamc@75 841 | false -> Unknown)
adamc@75 842 | false -> Unknown)))
adamc@75 843 | Bool b -> Found TBool
adamc@75 844 | And (e1, e2) ->
adamc@75 845 (match typeCheck e1 with
adamc@75 846 | Unknown -> Unknown
adamc@75 847 | Found t1 ->
adamc@75 848 (match typeCheck e2 with
adamc@75 849 | Unknown -> Unknown
adamc@75 850 | Found t2 ->
adamc@75 851 (match eq_type_dec t1 TBool with
adamc@75 852 | true ->
adamc@75 853 (match eq_type_dec t2 TBool with
adamc@75 854 | true -> Found TBool
adamc@75 855 | false -> Unknown)
adamc@75 856 | false -> Unknown)))
adamc@75 857 </pre># *)
adamc@75 858
adamc@75 859 (** %\smallskip%
adamc@75 860
adam@423 861 We can adapt this implementation to use [sumor], so that we know our type-checker only fails on ill-typed inputs. First, we define an analogue to the "assertion" notation. *)
adamc@73 862
adamc@77 863 (* begin thide *)
adamc@73 864 Notation "e1 ;;; e2" := (if e1 then e2 else !!)
adamc@73 865 (right associativity, at level 60).
adamc@73 866
adamc@75 867 (** Next, we prove a helpful lemma, which states that a given expression can have at most one type. *)
adamc@75 868
adamc@75 869 Lemma hasType_det : forall e t1,
adamc@73 870 hasType e t1
adam@335 871 -> forall t2, hasType e t2
adamc@73 872 -> t1 = t2.
adamc@73 873 induction 1; inversion 1; crush.
adamc@73 874 Qed.
adamc@73 875
adamc@75 876 (** Now we can define the type-checker. Its type expresses that it only fails on untypable expressions. *)
adamc@75 877
adamc@77 878 (* end thide *)
adam@297 879 Definition typeCheck' : forall e : exp, {t : type | hasType e t} + {forall t, ~ hasType e t}.
adamc@77 880 (* begin thide *)
adamc@73 881 Hint Constructors hasType.
adam@475 882
adamc@75 883 (** We register all of the typing rules as hints. *)
adamc@75 884
adamc@73 885 Hint Resolve hasType_det.
adam@475 886
adam@335 887 (** The lemma [hasType_det] will also be useful for proving proof obligations with contradictory contexts. Since its statement includes [forall]-bound variables that do not appear in its conclusion, only [eauto] will apply this hint. *)
adamc@73 888
adamc@75 889 (** Finally, the implementation of [typeCheck] can be transcribed literally, simply switching notations as needed. *)
adamc@212 890
adamc@212 891 refine (fix F (e : exp) : {t : type | hasType e t} + {forall t, ~ hasType e t} :=
adam@380 892 match e return {t : type | hasType e t} + {forall t, ~ hasType e t} with
adam@335 893 | Nat _ => [||TNat||]
adamc@73 894 | Plus e1 e2 =>
adamc@73 895 t1 <-- F e1;
adamc@73 896 t2 <-- F e2;
adamc@73 897 eq_type_dec t1 TNat;;;
adamc@73 898 eq_type_dec t2 TNat;;;
adam@335 899 [||TNat||]
adam@335 900 | Bool _ => [||TBool||]
adamc@73 901 | And e1 e2 =>
adamc@73 902 t1 <-- F e1;
adamc@73 903 t2 <-- F e2;
adamc@73 904 eq_type_dec t1 TBool;;;
adamc@73 905 eq_type_dec t2 TBool;;;
adam@335 906 [||TBool||]
adamc@73 907 end); clear F; crush' tt hasType; eauto.
adamc@75 908
adam@471 909 (** We clear [F], the local name for the recursive function, to avoid strange proofs that refer to recursive calls that we never make. Such a step is usually warranted when defining a recursive function with [refine]. The [crush] variant %\index{tactics!crush'}%[crush'] helps us by performing automatic inversion on instances of the predicates specified in its second argument. Once we throw in [eauto] to apply [hasType_det] for us, we have discharged all the subgoals. *)
adamc@77 910 (* end thide *)
adamc@212 911
adamc@212 912
adamc@73 913 Defined.
adamc@73 914
adamc@75 915 (** The short implementation here hides just how time-saving automation is. Every use of one of the notations adds a proof obligation, giving us 12 in total. Most of these obligations require multiple inversions and either uses of [hasType_det] or applications of [hasType] rules.
adamc@75 916
adam@335 917 Our new function remains easy to test: *)
adamc@75 918
adamc@73 919 Eval simpl in typeCheck' (Nat 0).
adamc@212 920 (** %\vspace{-.15in}% [[
adam@335 921 = [||TNat||]
adamc@75 922 : {t : type | hasType (Nat 0) t} +
adamc@75 923 {(forall t : type, ~ hasType (Nat 0) t)}
adam@302 924 ]]
adam@302 925 *)
adamc@75 926
adamc@73 927 Eval simpl in typeCheck' (Plus (Nat 1) (Nat 2)).
adamc@212 928 (** %\vspace{-.15in}% [[
adam@335 929 = [||TNat||]
adamc@75 930 : {t : type | hasType (Plus (Nat 1) (Nat 2)) t} +
adamc@75 931 {(forall t : type, ~ hasType (Plus (Nat 1) (Nat 2)) t)}
adam@302 932 ]]
adam@302 933 *)
adamc@75 934
adamc@73 935 Eval simpl in typeCheck' (Plus (Nat 1) (Bool false)).
adamc@212 936 (** %\vspace{-.15in}% [[
adamc@75 937 = !!
adamc@75 938 : {t : type | hasType (Plus (Nat 1) (Bool false)) t} +
adamc@75 939 {(forall t : type, ~ hasType (Plus (Nat 1) (Bool false)) t)}
adam@302 940 ]]
adam@335 941
adam@442 942 The results of simplifying calls to [typeCheck'] look deceptively similar to the results for [typeCheck], but now the types of the results provide more information. *)