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Proofreading pass through Chapter 16
author Adam Chlipala <adam@chlipala.net>
date Wed, 29 Aug 2012 16:33:59 -0400
parents 980962258b49
children 1fd4109f7b31
rev   line source
adam@381 1 (* Copyright (c) 2009-2012, Adam Chlipala
adamc@235 2 *
adamc@235 3 * This work is licensed under a
adamc@235 4 * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
adamc@235 5 * Unported License.
adamc@235 6 * The license text is available at:
adamc@235 7 * http://creativecommons.org/licenses/by-nc-nd/3.0/
adamc@235 8 *)
adamc@235 9
adamc@235 10 (* begin hide *)
adamc@236 11 Require Import Arith.
adamc@236 12
adam@314 13 Require Import CpdtTactics.
adamc@235 14
adamc@235 15 Set Implicit Arguments.
adamc@235 16 (* end hide *)
adamc@235 17
adamc@235 18
adam@381 19 (** %\part{The Big Picture}
adam@381 20
adam@381 21 \chapter{Proving in the Large}% *)
adamc@235 22
adam@433 23 (** It is somewhat unfortunate that the term "theorem proving" looks so much like the word "theory." Most researchers and practitioners in software assume that mechanized theorem proving is profoundly impractical. Indeed, until recently, most advances in theorem proving for higher-order logics have been largely theoretical. However, starting around the beginning of the 21st century, there was a surge in the use of proof assistants in serious verification efforts. That line of work is still quite new, but I believe it is not too soon to distill some lessons on how to work effectively with large formal proofs.
adamc@236 24
adamc@236 25 Thus, this chapter gives some tips for structuring and maintaining large Coq developments. *)
adamc@236 26
adamc@236 27
adamc@236 28 (** * Ltac Anti-Patterns *)
adamc@236 29
adam@433 30 (** In this book, I have been following an unusual style, where proofs are not considered finished until they are %\index{fully automated proofs}%"fully automated," in a certain sense. Each such theorem is proved by a single tactic. Since Ltac is a Turing-complete programming language, it is not hard to squeeze arbitrary heuristics into single tactics, using operators like the semicolon to combine steps. In contrast, most Ltac proofs "in the wild" consist of many steps, performed by individual tactics followed by periods. Is it really worth drawing a distinction between proof steps terminated by semicolons and steps terminated by periods?
adamc@236 31
adamc@237 32 I argue that this is, in fact, a very important distinction, with serious consequences for a majority of important verification domains. The more uninteresting drudge work a proof domain involves, the more important it is to work to prove theorems with single tactics. From an automation standpoint, single-tactic proofs can be extremely effective, and automation becomes more and more critical as proofs are populated by more uninteresting detail. In this section, I will give some examples of the consequences of more common proof styles.
adamc@236 33
adamc@236 34 As a running example, consider a basic language of arithmetic expressions, an interpreter for it, and a transformation that scales up every constant in an expression. *)
adamc@236 35
adamc@236 36 Inductive exp : Set :=
adamc@236 37 | Const : nat -> exp
adamc@236 38 | Plus : exp -> exp -> exp.
adamc@236 39
adamc@236 40 Fixpoint eval (e : exp) : nat :=
adamc@236 41 match e with
adamc@236 42 | Const n => n
adamc@236 43 | Plus e1 e2 => eval e1 + eval e2
adamc@236 44 end.
adamc@236 45
adamc@236 46 Fixpoint times (k : nat) (e : exp) : exp :=
adamc@236 47 match e with
adamc@236 48 | Const n => Const (k * n)
adamc@236 49 | Plus e1 e2 => Plus (times k e1) (times k e2)
adamc@236 50 end.
adamc@236 51
adam@452 52 (** We can write a very manual proof that [times] really implements multiplication. *)
adamc@236 53
adamc@236 54 Theorem eval_times : forall k e,
adamc@236 55 eval (times k e) = k * eval e.
adamc@236 56 induction e.
adamc@236 57
adamc@236 58 trivial.
adamc@236 59
adamc@236 60 simpl.
adamc@236 61 rewrite IHe1.
adamc@236 62 rewrite IHe2.
adamc@236 63 rewrite mult_plus_distr_l.
adamc@236 64 trivial.
adamc@236 65 Qed.
adamc@236 66
adam@368 67 (* begin thide *)
adam@367 68 (** We use spaces to separate the two inductive cases, but note that these spaces have no real semantic content; Coq does not enforce that our spacing matches the real case structure of a proof. The second case mentions automatically generated hypothesis names explicitly. As a result, innocuous changes to the theorem statement can invalidate the proof. *)
adamc@236 69
adamc@236 70 Reset eval_times.
adamc@236 71
adam@368 72 Theorem eval_times : forall k x,
adamc@236 73 eval (times k x) = k * eval x.
adamc@236 74 induction x.
adamc@236 75
adamc@236 76 trivial.
adamc@236 77
adamc@236 78 simpl.
adam@367 79 (** %\vspace{-.15in}%[[
adamc@236 80 rewrite IHe1.
adam@367 81 ]]
adamc@236 82
adam@367 83 <<
adamc@236 84 Error: The reference IHe1 was not found in the current environment.
adam@367 85 >>
adamc@236 86
adamc@236 87 The inductive hypotheses are named [IHx1] and [IHx2] now, not [IHe1] and [IHe2]. *)
adamc@236 88
adamc@236 89 Abort.
adamc@236 90
adamc@236 91 (** We might decide to use a more explicit invocation of [induction] to give explicit binders for all of the names that we will reference later in the proof. *)
adamc@236 92
adamc@236 93 Theorem eval_times : forall k e,
adamc@236 94 eval (times k e) = k * eval e.
adamc@236 95 induction e as [ | ? IHe1 ? IHe2 ].
adamc@236 96
adamc@236 97 trivial.
adamc@236 98
adamc@236 99 simpl.
adamc@236 100 rewrite IHe1.
adamc@236 101 rewrite IHe2.
adamc@236 102 rewrite mult_plus_distr_l.
adamc@236 103 trivial.
adamc@236 104 Qed.
adamc@236 105
adam@413 106 (** We pass %\index{tactics!induction}%[induction] an%\index{intro pattern}% _intro pattern_, using a [|] character to separate out instructions for the different inductive cases. Within a case, we write [?] to ask Coq to generate a name automatically, and we write an explicit name to assign that name to the corresponding new variable. It is apparent that, to use intro patterns to avoid proof brittleness, one needs to keep track of the seemingly unimportant facts of the orders in which variables are introduced. Thus, the script keeps working if we replace [e] by [x], but it has become more cluttered. Arguably, neither proof is particularly easy to follow.
adamc@236 107
adamc@237 108 That category of complaint has to do with understanding proofs as static artifacts. As with programming in general, with serious projects, it tends to be much more important to be able to support evolution of proofs as specifications change. Unstructured proofs like the above examples can be very hard to update in concert with theorem statements. For instance, consider how the last proof script plays out when we modify [times] to introduce a bug. *)
adamc@236 109
adamc@236 110 Reset times.
adamc@236 111
adamc@236 112 Fixpoint times (k : nat) (e : exp) : exp :=
adamc@236 113 match e with
adamc@236 114 | Const n => Const (1 + k * n)
adamc@236 115 | Plus e1 e2 => Plus (times k e1) (times k e2)
adamc@236 116 end.
adamc@236 117
adamc@236 118 Theorem eval_times : forall k e,
adamc@236 119 eval (times k e) = k * eval e.
adamc@236 120 induction e as [ | ? IHe1 ? IHe2 ].
adamc@236 121
adamc@236 122 trivial.
adamc@236 123
adamc@236 124 simpl.
adam@367 125 (** %\vspace{-.15in}%[[
adamc@236 126 rewrite IHe1.
adam@367 127 ]]
adamc@236 128
adam@367 129 <<
adamc@236 130 Error: The reference IHe1 was not found in the current environment.
adam@367 131 >>
adam@302 132 *)
adamc@236 133
adamc@236 134 Abort.
adamc@236 135
adam@398 136 (** Can you spot what went wrong, without stepping through the script step-by-step? The problem is that [trivial] never fails. Originally, [trivial] had been succeeding in proving an equality that follows by reflexivity. Our change to [times] leads to a case where that equality is no longer true. The invocation [trivial] happily leaves the false equality in place, and we continue on to the span of tactics intended for the second inductive case. Unfortunately, those tactics end up being applied to the _first_ case instead.
adamc@237 137
adam@433 138 The problem with [trivial] could be "solved" by writing, e.g., [solve [ trivial ]] instead, so that an error is signaled early on if something unexpected happens. However, the root problem is that the syntax of a tactic invocation does not imply how many subgoals it produces. Much more confusing instances of this problem are possible. For example, if a lemma [L] is modified to take an extra hypothesis, then uses of [apply L] will generate more subgoals than before. Old unstructured proof scripts will become hopelessly jumbled, with tactics applied to inappropriate subgoals. Because of the lack of structure, there is usually relatively little to be gleaned from knowledge of the precise point in a proof script where an error is raised. *)
adamc@236 139
adamc@236 140 Reset times.
adamc@236 141
adamc@236 142 Fixpoint times (k : nat) (e : exp) : exp :=
adamc@236 143 match e with
adamc@236 144 | Const n => Const (k * n)
adamc@236 145 | Plus e1 e2 => Plus (times k e1) (times k e2)
adamc@236 146 end.
adamc@236 147
adam@387 148 (** Many real developments try to make essentially unstructured proofs look structured by applying careful indentation conventions, idempotent case-marker tactics included solely to serve as documentation, and so on. All of these strategies suffer from the same kind of failure of abstraction that was just demonstrated. I like to say that if you find yourself caring about indentation in a proof script, it is a sign that the script is structured poorly.
adamc@236 149
adamc@236 150 We can rewrite the current proof with a single tactic. *)
adamc@236 151
adamc@236 152 Theorem eval_times : forall k e,
adamc@236 153 eval (times k e) = k * eval e.
adamc@236 154 induction e as [ | ? IHe1 ? IHe2 ]; [
adamc@236 155 trivial
adamc@236 156 | simpl; rewrite IHe1; rewrite IHe2; rewrite mult_plus_distr_l; trivial ].
adamc@236 157 Qed.
adamc@236 158
adam@462 159 (** We use the form of the semicolon operator that allows a different tactic to be specified for each generated subgoal. This is an improvement in robustness of the script. We no longer need to worry about tactics from one case being applied to a different case. Still, the proof script is not especially readable. Probably most readers would not find it helpful in explaining why the theorem is true. The same could be said for scripts which use the%\index{bullets}% _bullets_ or curly braces provided by Coq 8.4, which allow code like the above to be stepped through interactively, with periods in place of the semicolons, while representing proof structure in a way that is enforced by Coq. Interactive replay of scripts becomes easier, but readability is not really helped.
adamc@236 160
adamc@236 161 The situation gets worse in considering extensions to the theorem we want to prove. Let us add multiplication nodes to our [exp] type and see how the proof fares. *)
adamc@236 162
adamc@236 163 Reset exp.
adamc@236 164
adamc@236 165 Inductive exp : Set :=
adamc@236 166 | Const : nat -> exp
adamc@236 167 | Plus : exp -> exp -> exp
adamc@236 168 | Mult : exp -> exp -> exp.
adamc@236 169
adamc@236 170 Fixpoint eval (e : exp) : nat :=
adamc@236 171 match e with
adamc@236 172 | Const n => n
adamc@236 173 | Plus e1 e2 => eval e1 + eval e2
adamc@236 174 | Mult e1 e2 => eval e1 * eval e2
adamc@236 175 end.
adamc@236 176
adamc@236 177 Fixpoint times (k : nat) (e : exp) : exp :=
adamc@236 178 match e with
adamc@236 179 | Const n => Const (k * n)
adamc@236 180 | Plus e1 e2 => Plus (times k e1) (times k e2)
adamc@236 181 | Mult e1 e2 => Mult (times k e1) e2
adamc@236 182 end.
adamc@236 183
adamc@236 184 Theorem eval_times : forall k e,
adamc@236 185 eval (times k e) = k * eval e.
adam@367 186 (** %\vspace{-.25in}%[[
adamc@236 187 induction e as [ | ? IHe1 ? IHe2 ]; [
adamc@236 188 trivial
adamc@236 189 | simpl; rewrite IHe1; rewrite IHe2; rewrite mult_plus_distr_l; trivial ].
adam@367 190 ]]
adamc@236 191
adam@367 192 <<
adamc@236 193 Error: Expects a disjunctive pattern with 3 branches.
adam@367 194 >>
adam@302 195 *)
adamc@236 196 Abort.
adamc@236 197
adamc@236 198 (** Unsurprisingly, the old proof fails, because it explicitly says that there are two inductive cases. To update the script, we must, at a minimum, remember the order in which the inductive cases are generated, so that we can insert the new case in the appropriate place. Even then, it will be painful to add the case, because we cannot walk through proof steps interactively when they occur inside an explicit set of cases. *)
adamc@236 199
adamc@236 200 Theorem eval_times : forall k e,
adamc@236 201 eval (times k e) = k * eval e.
adamc@236 202 induction e as [ | ? IHe1 ? IHe2 | ? IHe1 ? IHe2 ]; [
adamc@236 203 trivial
adamc@236 204 | simpl; rewrite IHe1; rewrite IHe2; rewrite mult_plus_distr_l; trivial
adamc@236 205 | simpl; rewrite IHe1; rewrite mult_assoc; trivial ].
adamc@236 206 Qed.
adamc@236 207
adamc@236 208 (** Now we are in a position to see how much nicer is the style of proof that we have followed in most of this book. *)
adamc@236 209
adamc@236 210 Reset eval_times.
adamc@236 211
adam@375 212 Hint Rewrite mult_plus_distr_l.
adamc@238 213
adamc@236 214 Theorem eval_times : forall k e,
adamc@236 215 eval (times k e) = k * eval e.
adamc@236 216 induction e; crush.
adamc@236 217 Qed.
adam@368 218 (* end thide *)
adamc@236 219
adamc@237 220 (** This style is motivated by a hard truth: one person's manual proof script is almost always mostly inscrutable to most everyone else. I claim that step-by-step formal proofs are a poor way of conveying information. Thus, we had might as well cut out the steps and automate as much as possible.
adamc@237 221
adam@398 222 What about the illustrative value of proofs? Most informal proofs are read to convey the big ideas of proofs. How can reading [induction e; crush] convey any big ideas? My position is that any ideas that standard automation can find are not very big after all, and the _real_ big ideas should be expressed through lemmas that are added as hints.
adamc@237 223
adamc@237 224 An example should help illustrate what I mean. Consider this function, which rewrites an expression using associativity of addition and multiplication. *)
adamc@237 225
adamc@237 226 Fixpoint reassoc (e : exp) : exp :=
adamc@237 227 match e with
adamc@237 228 | Const _ => e
adamc@237 229 | Plus e1 e2 =>
adamc@237 230 let e1' := reassoc e1 in
adamc@237 231 let e2' := reassoc e2 in
adamc@237 232 match e2' with
adamc@237 233 | Plus e21 e22 => Plus (Plus e1' e21) e22
adamc@237 234 | _ => Plus e1' e2'
adamc@237 235 end
adamc@237 236 | Mult e1 e2 =>
adamc@237 237 let e1' := reassoc e1 in
adamc@237 238 let e2' := reassoc e2 in
adamc@237 239 match e2' with
adamc@237 240 | Mult e21 e22 => Mult (Mult e1' e21) e22
adamc@237 241 | _ => Mult e1' e2'
adamc@237 242 end
adamc@237 243 end.
adamc@237 244
adamc@237 245 Theorem reassoc_correct : forall e, eval (reassoc e) = eval e.
adam@368 246 (* begin thide *)
adamc@237 247 induction e; crush;
adamc@237 248 match goal with
adam@413 249 | [ |- context[match ?E with Const _ => _ | _ => _ end] ] =>
adamc@237 250 destruct E; crush
adamc@237 251 end.
adamc@237 252
adamc@237 253 (** One subgoal remains:
adamc@237 254 [[
adamc@237 255 IHe2 : eval e3 * eval e4 = eval e2
adamc@237 256 ============================
adamc@237 257 eval e1 * eval e3 * eval e4 = eval e1 * eval e2
adamc@237 258 ]]
adamc@237 259
adam@433 260 The [crush] tactic does not know how to finish this goal. We could finish the proof manually. *)
adamc@237 261
adamc@237 262 rewrite <- IHe2; crush.
adamc@237 263
adamc@237 264 (** However, the proof would be easier to understand and maintain if we separated this insight into a separate lemma. *)
adamc@237 265
adamc@237 266 Abort.
adamc@237 267
adamc@237 268 Lemma rewr : forall a b c d, b * c = d -> a * b * c = a * d.
adamc@237 269 crush.
adamc@237 270 Qed.
adamc@237 271
adamc@237 272 Hint Resolve rewr.
adamc@237 273
adamc@237 274 Theorem reassoc_correct : forall e, eval (reassoc e) = eval e.
adamc@237 275 induction e; crush;
adamc@237 276 match goal with
adam@413 277 | [ |- context[match ?E with Const _ => _ | _ => _ end] ] =>
adamc@237 278 destruct E; crush
adamc@237 279 end.
adamc@237 280 Qed.
adam@368 281 (* end thide *)
adamc@237 282
adamc@237 283 (** In the limit, a complicated inductive proof might rely on one hint for each inductive case. The lemma for each hint could restate the associated case. Compared to manual proof scripts, we arrive at more readable results. Scripts no longer need to depend on the order in which cases are generated. The lemmas are easier to digest separately than are fragments of tactic code, since lemma statements include complete proof contexts. Such contexts can only be extracted from monolithic manual proofs by stepping through scripts interactively.
adamc@237 284
adam@387 285 The more common situation is that a large induction has several easy cases that automation makes short work of. In the remaining cases, automation performs some standard simplification. Among these cases, some may require quite involved proofs; such a case may deserve a hint lemma of its own, where the lemma statement may copy the simplified version of the case. Alternatively, the proof script for the main theorem may be extended with some automation code targeted at the specific case. Even such targeted scripting is more desirable than manual proving, because it may be read and understood without knowledge of a proof's hierarchical structure, case ordering, or name binding structure.
adam@387 286
adam@413 287 A competing alternative to the common style of Coq tactics is the%\index{declarative proof scripts}% _declarative_ style, most frequently associated today with the %\index{Isar}%Isar%~\cite{Isar}% language. A declarative proof script is very explicit about subgoal structure and introduction of local names, aiming for human readability. The coding of proof automation is taken to be outside the scope of the proof language, an assumption related to the idea that it is not worth building new automation for each serious theorem. I have shown in this book many examples of theorem-specific automation, which I believe is crucial for scaling to significant results. Declarative proof scripts make it easier to read scripts to modify them for theorem statement changes, but the alternate%\index{adaptive proof scripts}% _adaptive_ style from this book allows use of the _same_ scripts for many versions of a theorem.
adam@387 288
adam@387 289 Perhaps I am a pessimist for thinking that fully formal proofs will inevitably consist of details that are uninteresting to people, but it is my preference to focus on conveying proof-specific details through choice of lemmas. Additionally, adaptive Ltac scripts contain bits of automation that can be understood in isolation. For instance, in a big [repeat match] loop, each case can generally be digested separately, which is a big contrast from trying to understand the hierarchical structure of a script in a more common style. Adaptive scripts rely on variable binding, but generally only over very small scopes, whereas understanding a traditional script requires tracking the identities of local variables potentially across pages of code.
adam@387 290
adam@398 291 One might also wonder why it makes sense to prove all theorems automatically (in the sense of adaptive proof scripts) but not construct all programs automatically. My view there is that _program synthesis_ is a very useful idea that deserves broader application! In practice, there are difficult obstacles in the way of finding a program automatically from its specification. A typical specification is not exhaustive in its description of program properties. For instance, details of performance on particular machine architectures are often omitted. As a result, a synthesized program may be correct in some sense while suffering from deficiencies in other senses. Program synthesis research will continue to come up with ways of dealing with this problem, but the situation for theorem proving is fundamentally different. Following mathematical practice, the only property of a formal proof that we care about is which theorem it proves, and it is trivial to check this property automatically. In other words, with a simple criterion for what makes a proof acceptable, automatic search is straightforward. Of course, in practice we also care about understandability of proofs to facilitate long-term maintenance, and that is just what the techniques outlined above are meant to support, and the next section gives some related advice. *)
adamc@237 292
adamc@235 293
adamc@238 294 (** * Debugging and Maintaining Automation *)
adamc@238 295
adam@367 296 (** Fully automated proofs are desirable because they open up possibilities for automatic adaptation to changes of specification. A well-engineered script within a narrow domain can survive many changes to the formulation of the problem it solves. Still, as we are working with higher-order logic, most theorems fall within no obvious decidable theories. It is inevitable that most long-lived automated proofs will need updating.
adamc@238 297
adam@367 298 Before we are ready to update our proofs, we need to write them in the first place. While fully automated scripts are most robust to changes of specification, it is hard to write every new proof directly in that form. Instead, it is useful to begin a theorem with exploratory proving and then gradually refine it into a suitable automated form.
adamc@238 299
adam@387 300 Consider this theorem from Chapter 8, which we begin by proving in a mostly manual way, invoking [crush] after each step to discharge any low-hanging fruit. Our manual effort involves choosing which expressions to case-analyze on. *)
adamc@238 301
adamc@238 302 (* begin hide *)
adamc@238 303 Require Import MoreDep.
adamc@238 304 (* end hide *)
adamc@238 305
adamc@238 306 Theorem cfold_correct : forall t (e : exp t), expDenote e = expDenote (cfold e).
adam@368 307 (* begin thide *)
adamc@238 308 induction e; crush.
adamc@238 309
adamc@238 310 dep_destruct (cfold e1); crush.
adamc@238 311 dep_destruct (cfold e2); crush.
adamc@238 312
adamc@238 313 dep_destruct (cfold e1); crush.
adamc@238 314 dep_destruct (cfold e2); crush.
adamc@238 315
adamc@238 316 dep_destruct (cfold e1); crush.
adamc@238 317 dep_destruct (cfold e2); crush.
adamc@238 318
adamc@238 319 dep_destruct (cfold e1); crush.
adamc@238 320 dep_destruct (expDenote e1); crush.
adamc@238 321
adamc@238 322 dep_destruct (cfold e); crush.
adamc@238 323
adamc@238 324 dep_destruct (cfold e); crush.
adamc@238 325 Qed.
adamc@238 326
adamc@238 327 (** In this complete proof, it is hard to avoid noticing a pattern. We rework the proof, abstracting over the patterns we find. *)
adamc@238 328
adamc@238 329 Reset cfold_correct.
adamc@238 330
adamc@238 331 Theorem cfold_correct : forall t (e : exp t), expDenote e = expDenote (cfold e).
adamc@238 332 induction e; crush.
adamc@238 333
adamc@238 334 (** The expression we want to destruct here turns out to be the discriminee of a [match], and we can easily enough write a tactic that destructs all such expressions. *)
adamc@238 335
adamc@238 336 Ltac t :=
adamc@238 337 repeat (match goal with
adam@413 338 | [ |- context[match ?E with NConst _ => _ | _ => _ end] ] =>
adamc@238 339 dep_destruct E
adamc@238 340 end; crush).
adamc@238 341
adamc@238 342 t.
adamc@238 343
adamc@238 344 (** This tactic invocation discharges the whole case. It does the same on the next two cases, but it gets stuck on the fourth case. *)
adamc@238 345
adamc@238 346 t.
adamc@238 347
adamc@238 348 t.
adamc@238 349
adamc@238 350 t.
adamc@238 351
adamc@238 352 (** The subgoal's conclusion is:
adamc@238 353 [[
adamc@238 354 ============================
adamc@238 355 (if expDenote e1 then expDenote (cfold e2) else expDenote (cfold e3)) =
adamc@238 356 expDenote (if expDenote e1 then cfold e2 else cfold e3)
adamc@238 357 ]]
adamc@238 358
adamc@238 359 We need to expand our [t] tactic to handle this case. *)
adamc@238 360
adamc@238 361 Ltac t' :=
adamc@238 362 repeat (match goal with
adam@413 363 | [ |- context[match ?E with NConst _ => _ | _ => _ end] ] =>
adamc@238 364 dep_destruct E
adamc@238 365 | [ |- (if ?E then _ else _) = _ ] => destruct E
adamc@238 366 end; crush).
adamc@238 367
adamc@238 368 t'.
adamc@238 369
adamc@238 370 (** Now the goal is discharged, but [t'] has no effect on the next subgoal. *)
adamc@238 371
adamc@238 372 t'.
adamc@238 373
adamc@238 374 (** A final revision of [t] finishes the proof. *)
adamc@238 375
adamc@238 376 Ltac t'' :=
adamc@238 377 repeat (match goal with
adam@413 378 | [ |- context[match ?E with NConst _ => _ | _ => _ end] ] =>
adamc@238 379 dep_destruct E
adamc@238 380 | [ |- (if ?E then _ else _) = _ ] => destruct E
adamc@238 381 | [ |- context[match pairOut ?E with Some _ => _
adamc@238 382 | None => _ end] ] =>
adamc@238 383 dep_destruct E
adamc@238 384 end; crush).
adamc@238 385
adamc@238 386 t''.
adamc@238 387
adamc@238 388 t''.
adamc@238 389 Qed.
adamc@238 390
adam@367 391 (** We can take the final tactic and move it into the initial part of the proof script, arriving at a nicely automated proof. *)
adamc@238 392
adamc@238 393 Reset t.
adamc@238 394
adamc@238 395 Theorem cfold_correct : forall t (e : exp t), expDenote e = expDenote (cfold e).
adamc@238 396 induction e; crush;
adamc@238 397 repeat (match goal with
adam@413 398 | [ |- context[match ?E with NConst _ => _ | _ => _ end] ] =>
adamc@238 399 dep_destruct E
adamc@238 400 | [ |- (if ?E then _ else _) = _ ] => destruct E
adamc@238 401 | [ |- context[match pairOut ?E with Some _ => _
adamc@238 402 | None => _ end] ] =>
adamc@238 403 dep_destruct E
adamc@238 404 end; crush).
adamc@238 405 Qed.
adam@368 406 (* end thide *)
adamc@238 407
adam@367 408 (** Even after we put together nice automated proofs, we must deal with specification changes that can invalidate them. It is not generally possible to step through single-tactic proofs interactively. There is a command %\index{Vernacular commands!Debug On}%[Debug On] that lets us step through points in tactic execution, but the debugger tends to make counterintuitive choices of which points we would like to stop at, and per-point output is quite verbose, so most Coq users do not find this debugging mode very helpful. How are we to understand what has broken in a script that used to work?
adamc@240 409
adamc@240 410 An example helps demonstrate a useful approach. Consider what would have happened in our proof of [reassoc_correct] if we had first added an unfortunate rewriting hint. *)
adamc@240 411
adamc@240 412 Reset reassoc_correct.
adamc@240 413
adamc@240 414 Theorem confounder : forall e1 e2 e3,
adamc@240 415 eval e1 * eval e2 * eval e3 = eval e1 * (eval e2 + 1 - 1) * eval e3.
adamc@240 416 crush.
adamc@240 417 Qed.
adamc@240 418
adam@375 419 Hint Rewrite confounder.
adamc@240 420
adamc@240 421 Theorem reassoc_correct : forall e, eval (reassoc e) = eval e.
adam@368 422 (* begin thide *)
adamc@240 423 induction e; crush;
adamc@240 424 match goal with
adam@413 425 | [ |- context[match ?E with Const _ => _ | _ => _ end] ] =>
adamc@240 426 destruct E; crush
adamc@240 427 end.
adamc@240 428
adamc@240 429 (** One subgoal remains:
adamc@240 430
adamc@240 431 [[
adamc@240 432 ============================
adamc@240 433 eval e1 * (eval e3 + 1 - 1) * eval e4 = eval e1 * eval e2
adamc@240 434 ]]
adamc@240 435
adam@367 436 The poorly chosen rewrite rule fired, changing the goal to a form where another hint no longer applies. Imagine that we are in the middle of a large development with many hints. How would we diagnose the problem? First, we might not be sure which case of the inductive proof has gone wrong. It is useful to separate out our automation procedure and apply it manually. *)
adamc@240 437
adamc@240 438 Restart.
adamc@240 439
adamc@240 440 Ltac t := crush; match goal with
adam@413 441 | [ |- context[match ?E with Const _ => _ | _ => _ end] ] =>
adamc@240 442 destruct E; crush
adamc@240 443 end.
adamc@240 444
adamc@240 445 induction e.
adamc@240 446
adam@387 447 (** Since we see the subgoals before any simplification occurs, it is clear that this is the case for constants. Our [t] makes short work of it. *)
adamc@240 448
adamc@240 449 t.
adamc@240 450
adamc@240 451 (** The next subgoal, for addition, is also discharged without trouble. *)
adamc@240 452
adamc@240 453 t.
adamc@240 454
adamc@240 455 (** The final subgoal is for multiplication, and it is here that we get stuck in the proof state summarized above. *)
adamc@240 456
adamc@240 457 t.
adamc@240 458
adam@433 459 (** What is [t] doing to get us to this point? The %\index{tactics!info}%[info] command can help us answer this kind of question. (As of this writing, [info] is no longer functioning in the most recent Coq release, but I hope it returns.) *)
adamc@240 460
adamc@240 461 Undo.
adamc@240 462 info t.
adam@413 463
adam@433 464 (* begin hide *)
adam@437 465 (* begin thide *)
adam@433 466 Definition eir := eq_ind_r.
adam@437 467 (* end thide *)
adam@433 468 (* end hide *)
adam@433 469
adam@367 470 (** %\vspace{-.15in}%[[
adam@375 471 == simpl in *; intuition; subst; autorewrite with core in *;
adam@375 472 simpl in *; intuition; subst; autorewrite with core in *;
adamc@240 473 simpl in *; intuition; subst; destruct (reassoc e2).
adamc@240 474 simpl in *; intuition.
adamc@240 475
adamc@240 476 simpl in *; intuition.
adamc@240 477
adam@375 478 simpl in *; intuition; subst; autorewrite with core in *;
adamc@240 479 refine (eq_ind_r
adamc@240 480 (fun n : nat =>
adamc@240 481 n * (eval e3 + 1 - 1) * eval e4 = eval e1 * eval e2) _ IHe1);
adam@375 482 autorewrite with core in *; simpl in *; intuition;
adam@375 483 subst; autorewrite with core in *; simpl in *;
adamc@240 484 intuition; subst.
adamc@240 485
adamc@240 486 ]]
adamc@240 487
adamc@240 488 A detailed trace of [t]'s execution appears. Since we are using the very general [crush] tactic, many of these steps have no effect and only occur as instances of a more general strategy. We can copy-and-paste the details to see where things go wrong. *)
adamc@240 489
adamc@240 490 Undo.
adamc@240 491
adamc@240 492 (** We arbitrarily split the script into chunks. The first few seem not to do any harm. *)
adamc@240 493
adam@375 494 simpl in *; intuition; subst; autorewrite with core in *.
adam@375 495 simpl in *; intuition; subst; autorewrite with core in *.
adamc@240 496 simpl in *; intuition; subst; destruct (reassoc e2).
adamc@240 497 simpl in *; intuition.
adamc@240 498 simpl in *; intuition.
adamc@240 499
adamc@240 500 (** The next step is revealed as the culprit, bringing us to the final unproved subgoal. *)
adamc@240 501
adam@375 502 simpl in *; intuition; subst; autorewrite with core in *.
adamc@240 503
adamc@240 504 (** We can split the steps further to assign blame. *)
adamc@240 505
adamc@240 506 Undo.
adamc@240 507
adamc@240 508 simpl in *.
adamc@240 509 intuition.
adamc@240 510 subst.
adam@375 511 autorewrite with core in *.
adamc@240 512
adamc@240 513 (** It was the final of these four tactics that made the rewrite. We can find out exactly what happened. The [info] command presents hierarchical views of proof steps, and we can zoom down to a lower level of detail by applying [info] to one of the steps that appeared in the original trace. *)
adamc@240 514
adamc@240 515 Undo.
adamc@240 516
adam@375 517 info autorewrite with core in *.
adam@367 518 (** %\vspace{-.15in}%[[
adamc@240 519 == refine (eq_ind_r (fun n : nat => n = eval e1 * eval e2) _
adamc@240 520 (confounder (reassoc e1) e3 e4)).
adamc@240 521 ]]
adamc@240 522
adamc@240 523 The way a rewrite is displayed is somewhat baroque, but we can see that theorem [confounder] is the final culprit. At this point, we could remove that hint, prove an alternate version of the key lemma [rewr], or come up with some other remedy. Fixing this kind of problem tends to be relatively easy once the problem is revealed. *)
adamc@240 524
adamc@240 525 Abort.
adam@368 526 (* end thide *)
adamc@240 527
adamc@241 528 (** Sometimes a change to a development has undesirable performance consequences, even if it does not prevent any old proof scripts from completing. If the performance consequences are severe enough, the proof scripts can be considered broken for practical purposes.
adamc@241 529
adamc@241 530 Here is one example of a performance surprise. *)
adamc@241 531
adamc@239 532 Section slow.
adamc@239 533 Hint Resolve trans_eq.
adamc@239 534
adamc@241 535 (** The central element of the problem is the addition of transitivity as a hint. With transitivity available, it is easy for proof search to wind up exploring exponential search spaces. We also add a few other arbitrary variables and hypotheses, designed to lead to trouble later. *)
adamc@241 536
adamc@239 537 Variable A : Set.
adamc@239 538 Variables P Q R S : A -> A -> Prop.
adamc@239 539 Variable f : A -> A.
adamc@239 540
adamc@239 541 Hypothesis H1 : forall x y, P x y -> Q x y -> R x y -> f x = f y.
adamc@239 542 Hypothesis H2 : forall x y, S x y -> R x y.
adamc@239 543
adam@367 544 (** We prove a simple lemma very quickly, using the %\index{Vernacular commands!Time}%[Time] command to measure exactly how quickly. *)
adamc@241 545
adamc@239 546 Lemma slow : forall x y, P x y -> Q x y -> S x y -> f x = f y.
adamc@241 547 Time eauto 6.
adam@433 548 (** <<
adamc@241 549 Finished transaction in 0. secs (0.068004u,0.s)
adam@433 550 >>
adam@302 551 *)
adamc@241 552
adamc@239 553 Qed.
adamc@239 554
adamc@241 555 (** Now we add a different hypothesis, which is innocent enough; in fact, it is even provable as a theorem. *)
adamc@241 556
adamc@239 557 Hypothesis H3 : forall x y, x = y -> f x = f y.
adamc@239 558
adamc@239 559 Lemma slow' : forall x y, P x y -> Q x y -> S x y -> f x = f y.
adamc@241 560 Time eauto 6.
adam@433 561 (** <<
adamc@241 562 Finished transaction in 2. secs (1.264079u,0.s)
adam@433 563 >>
adam@445 564 %\vspace{-.15in}%Why has the search time gone up so much? The [info] command is not much help, since it only shows the result of search, not all of the paths that turned out to be worthless. *)
adamc@241 565
adam@368 566 (* begin thide *)
adamc@241 567 Restart.
adamc@241 568 info eauto 6.
adam@367 569 (** %\vspace{-.15in}%[[
adamc@241 570 == intro x; intro y; intro H; intro H0; intro H4;
adamc@241 571 simple eapply trans_eq.
adam@426 572 simple apply eq_refl.
adamc@241 573
adamc@241 574 simple eapply trans_eq.
adam@426 575 simple apply eq_refl.
adamc@241 576
adamc@241 577 simple eapply trans_eq.
adam@426 578 simple apply eq_refl.
adamc@241 579
adamc@241 580 simple apply H1.
adamc@241 581 eexact H.
adamc@241 582
adamc@241 583 eexact H0.
adamc@241 584
adamc@241 585 simple apply H2; eexact H4.
adamc@241 586 ]]
adamc@241 587
adam@367 588 This output does not tell us why proof search takes so long, but it does provide a clue that would be useful if we had forgotten that we added transitivity as a hint. The [eauto] tactic is applying depth-first search, and the proof script where the real action is ends up buried inside a chain of pointless invocations of transitivity, where each invocation uses reflexivity to discharge one subgoal. Each increment to the depth argument to [eauto] adds another silly use of transitivity. This wasted proof effort only adds linear time overhead, as long as proof search never makes false steps. No false steps were made before we added the new hypothesis, but somehow the addition made possible a new faulty path. To understand which paths we enabled, we can use the %\index{tactics!debug}%[debug] command. *)
adamc@241 589
adamc@241 590 Restart.
adamc@241 591 debug eauto 6.
adamc@241 592
adam@433 593 (* begin hide *)
adam@437 594 (* begin thide *)
adam@433 595 Definition deeeebug := (@eq_refl, @sym_eq).
adam@437 596 (* end thide *)
adam@433 597 (* end hide *)
adam@433 598
adamc@241 599 (** The output is a large proof tree. The beginning of the tree is enough to reveal what is happening:
adamc@241 600 [[
adamc@241 601 1 depth=6
adamc@241 602 1.1 depth=6 intro
adamc@241 603 1.1.1 depth=6 intro
adamc@241 604 1.1.1.1 depth=6 intro
adamc@241 605 1.1.1.1.1 depth=6 intro
adamc@241 606 1.1.1.1.1.1 depth=6 intro
adamc@241 607 1.1.1.1.1.1.1 depth=5 apply H3
adamc@241 608 1.1.1.1.1.1.1.1 depth=4 eapply trans_eq
adam@426 609 1.1.1.1.1.1.1.1.1 depth=4 apply eq_refl
adamc@241 610 1.1.1.1.1.1.1.1.1.1 depth=3 eapply trans_eq
adam@426 611 1.1.1.1.1.1.1.1.1.1.1 depth=3 apply eq_refl
adamc@241 612 1.1.1.1.1.1.1.1.1.1.1.1 depth=2 eapply trans_eq
adam@426 613 1.1.1.1.1.1.1.1.1.1.1.1.1 depth=2 apply eq_refl
adamc@241 614 1.1.1.1.1.1.1.1.1.1.1.1.1.1 depth=1 eapply trans_eq
adam@426 615 1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 depth=1 apply eq_refl
adamc@241 616 1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 depth=0 eapply trans_eq
adamc@241 617 1.1.1.1.1.1.1.1.1.1.1.1.1.1.2 depth=1 apply sym_eq ; trivial
adamc@241 618 1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1 depth=0 eapply trans_eq
adamc@241 619 1.1.1.1.1.1.1.1.1.1.1.1.1.1.3 depth=0 eapply trans_eq
adamc@241 620 1.1.1.1.1.1.1.1.1.1.1.1.2 depth=2 apply sym_eq ; trivial
adamc@241 621 1.1.1.1.1.1.1.1.1.1.1.1.2.1 depth=1 eapply trans_eq
adam@426 622 1.1.1.1.1.1.1.1.1.1.1.1.2.1.1 depth=1 apply eq_refl
adamc@241 623 1.1.1.1.1.1.1.1.1.1.1.1.2.1.1.1 depth=0 eapply trans_eq
adamc@241 624 1.1.1.1.1.1.1.1.1.1.1.1.2.1.2 depth=1 apply sym_eq ; trivial
adamc@241 625 1.1.1.1.1.1.1.1.1.1.1.1.2.1.2.1 depth=0 eapply trans_eq
adamc@241 626 1.1.1.1.1.1.1.1.1.1.1.1.2.1.3 depth=0 eapply trans_eq
adamc@241 627 ]]
adamc@241 628
adam@367 629 The first choice [eauto] makes is to apply [H3], since [H3] has the fewest hypotheses of all of the hypotheses and hints that match. However, it turns out that the single hypothesis generated is unprovable. That does not stop [eauto] from trying to prove it with an exponentially sized tree of applications of transitivity, reflexivity, and symmetry of equality. It is the children of the initial [apply H3] that account for all of the noticeable time in proof execution. In a more realistic development, we might use this output of [debug] to realize that adding transitivity as a hint was a bad idea. *)
adamc@241 630
adamc@239 631 Qed.
adam@368 632 (* end thide *)
adamc@239 633 End slow.
adamc@239 634
adam@387 635 (** As aggravating as the above situation may be, there is greater aggravation to be had from importing library modules with commands like %\index{Vernacular commands!Require Import}%[Require Import]. Such a command imports not just the Gallina terms from a module, but also all the hints for [auto], [eauto], and [autorewrite]. Some very recent versions of Coq include mechanisms for removing hints from databases, but the proper solution is to be very conservative in exporting hints from modules. Consider putting hints in named databases, so that they may be used only when called upon explicitly, as demonstrated in Chapter 13.
adam@387 636
adam@413 637 It is also easy to end up with a proof script that uses too much memory. As tactics run, they avoid generating proof terms, since serious proof search will consider many possible avenues, and we do not want to build proof terms for subproofs that end up unused. Instead, tactic execution maintains%\index{thunks}% _thunks_ (suspended computations, represented with closures), such that a tactic's proof-producing thunk is only executed when we run %\index{Vernacular commands!Qed}%[Qed]. These thunks can use up large amounts of space, such that a proof script exhausts available memory, even when we know that we could have used much less memory by forcing some thunks earlier.
adamc@241 638
adam@433 639 The %\index{tactics!abstract}%[abstract] tactical helps us force thunks by proving some subgoals as their own lemmas. For instance, a proof [induction x; crush] can in many cases be made to use significantly less peak memory by changing it to [induction x; abstract crush]. The main limitation of [abstract] is that it can only be applied to subgoals that are proved completely, with no undetermined unification variables in their initial states. Still, many large automated proofs can realize vast memory savings via [abstract]. *)
adamc@241 640
adamc@238 641
adamc@235 642 (** * Modules *)
adamc@235 643
adam@462 644 (** Last chapter's examples of proof by reflection demonstrate opportunities for implementing abstract proof strategies with stronger formal guarantees than can be had with Ltac scripting. Coq's _module system_ provides another tool for more rigorous development of generic theorems. This feature is inspired by the module systems found in Standard ML%~\cite{modules}% and OCaml, and the discussion that follows assumes familiarity with the basics of one of those systems.
adamc@242 645
adam@413 646 ML modules facilitate the grouping of %\index{abstract type}%abstract types with operations over those types. Moreover, there is support for%\index{functor}% _functors_, which are functions from modules to modules. A canonical example of a functor is one that builds a data structure implementation from a module that describes a domain of keys and its associated comparison operations.
adamc@242 647
adam@462 648 When we add modules to a base language with dependent types, it becomes possible to use modules and functors to formalize kinds of reasoning that are common in algebra. For instance, the following module signature captures the essence of the algebraic structure known as a group. A group consists of a carrier set [G], an associative binary operation [f], a left identity element [id] for [f], and an operation [i] that is a left inverse for [f].%\index{Vernacular commands!Module Type}% *)
adamc@242 649
adamc@235 650 Module Type GROUP.
adamc@235 651 Parameter G : Set.
adamc@235 652 Parameter f : G -> G -> G.
adam@462 653 Parameter id : G.
adamc@235 654 Parameter i : G -> G.
adamc@235 655
adamc@235 656 Axiom assoc : forall a b c, f (f a b) c = f a (f b c).
adam@462 657 Axiom ident : forall a, f id a = a.
adam@462 658 Axiom inverse : forall a, f (i a) a = id.
adamc@235 659 End GROUP.
adamc@235 660
adam@367 661 (** Many useful theorems hold of arbitrary groups. We capture some such theorem statements in another module signature.%\index{Vernacular commands!Declare Module}% *)
adamc@242 662
adamc@235 663 Module Type GROUP_THEOREMS.
adamc@235 664 Declare Module M : GROUP.
adamc@235 665
adam@462 666 Axiom ident' : forall a, M.f a M.id = a.
adamc@235 667
adam@462 668 Axiom inverse' : forall a, M.f a (M.i a) = M.id.
adamc@235 669
adam@462 670 Axiom unique_ident : forall id', (forall a, M.f id' a = a) -> id' = M.id.
adamc@235 671 End GROUP_THEOREMS.
adamc@235 672
adam@387 673 (** We implement generic proofs of these theorems with a functor, whose input is an arbitrary group [M]. %\index{Vernacular commands!Module}% *)
adamc@242 674
adam@387 675 Module GroupProofs (M : GROUP) : GROUP_THEOREMS with Module M := M.
adam@413 676 (** As in ML, Coq provides multiple options for ascribing signatures to modules. Here we use just the colon operator, which implements%\index{opaque ascription}% _opaque ascription_, hiding all details of the module not exposed by the signature. Another option is%\index{transparent ascription}% _transparent ascription_ via the [<:] operator, which checks for signature compatibility without hiding implementation details. Here we stick with opaque ascription but employ the [with] operation to add more detail to a signature, exposing just those implementation details that we need to. For instance, here we expose the underlying group representation set and operator definitions. Without such a refinement, we would get an output module proving theorems about some unknown group, which is not very useful. Also note that opaque ascription can in Coq have some undesirable consequences without analogues in ML, since not just the types but also the _definitions_ of identifiers have significance in type checking and theorem proving. *)
adam@387 677
adamc@235 678 Module M := M.
adam@387 679 (** To ensure that the module we are building meets the [GROUP_THEOREMS] signature, we add an extra local name for [M], the functor argument. *)
adamc@235 680
adamc@235 681 Import M.
adam@387 682 (** It would be inconvenient to repeat the prefix [M.] everywhere in our theorem statements and proofs, so we bring all the identifiers of [M] into the local scope unqualified.
adam@387 683
adam@387 684 Now we are ready to prove the three theorems. The proofs are completely manual, which may seem ironic given the content of the previous sections! This illustrates another lesson, which is that short proof scripts that change infrequently may be worth leaving unautomated. It would take some effort to build suitable generic automation for these theorems about groups, so I stick with manual proof scripts to avoid distracting us from the main message of the section. We take the proofs from the Wikipedia page on elementary group theory. *)
adamc@235 685
adam@462 686 Theorem inverse' : forall a, f a (i a) = id.
adamc@235 687 intro.
adamc@235 688 rewrite <- (ident (f a (i a))).
adamc@235 689 rewrite <- (inverse (f a (i a))) at 1.
adamc@235 690 rewrite assoc.
adamc@235 691 rewrite assoc.
adamc@235 692 rewrite <- (assoc (i a) a (i a)).
adamc@235 693 rewrite inverse.
adamc@235 694 rewrite ident.
adamc@235 695 apply inverse.
adamc@235 696 Qed.
adamc@235 697
adam@462 698 Theorem ident' : forall a, f a id = a.
adamc@235 699 intro.
adamc@235 700 rewrite <- (inverse a).
adamc@235 701 rewrite <- assoc.
adamc@235 702 rewrite inverse'.
adamc@235 703 apply ident.
adamc@235 704 Qed.
adamc@235 705
adam@462 706 Theorem unique_ident : forall id', (forall a, M.f id' a = a) -> id' = M.id.
adamc@235 707 intros.
adam@462 708 rewrite <- (H id).
adamc@235 709 symmetry.
adamc@235 710 apply ident'.
adamc@235 711 Qed.
adam@387 712 End GroupProofs.
adamc@239 713
adamc@242 714 (** We can show that the integers with [+] form a group. *)
adamc@242 715
adamc@239 716 Require Import ZArith.
adamc@239 717 Open Scope Z_scope.
adamc@239 718
adamc@239 719 Module Int.
adamc@239 720 Definition G := Z.
adamc@239 721 Definition f x y := x + y.
adam@462 722 Definition id := 0.
adamc@239 723 Definition i x := -x.
adamc@239 724
adamc@239 725 Theorem assoc : forall a b c, f (f a b) c = f a (f b c).
adamc@239 726 unfold f; crush.
adamc@239 727 Qed.
adam@462 728 Theorem ident : forall a, f id a = a.
adam@462 729 unfold f, id; crush.
adamc@239 730 Qed.
adam@462 731 Theorem inverse : forall a, f (i a) a = id.
adam@462 732 unfold f, i, id; crush.
adamc@239 733 Qed.
adamc@239 734 End Int.
adamc@239 735
adamc@242 736 (** Next, we can produce integer-specific versions of the generic group theorems. *)
adamc@242 737
adam@387 738 Module IntProofs := GroupProofs(Int).
adamc@239 739
adam@387 740 Check IntProofs.unique_ident.
adamc@242 741 (** %\vspace{-.15in}% [[
adam@387 742 IntProofs.unique_ident
adamc@242 743 : forall e' : Int.G, (forall a : Int.G, Int.f e' a = a) -> e' = Int.e
adam@302 744 ]]
adam@367 745
adam@367 746 Projections like [Int.G] are known to be definitionally equal to the concrete values we have assigned to them, so the above theorem yields as a trivial corollary the following more natural restatement: *)
adamc@239 747
adam@462 748 Theorem unique_ident : forall id', (forall a, id' + a = a) -> id' = 0.
adam@368 749 (* begin thide *)
adam@387 750 exact IntProofs.unique_ident.
adamc@239 751 Qed.
adam@368 752 (* end thide *)
adamc@242 753
adam@462 754 (** As in ML, the module system provides an effective way to structure large developments. Unlike in ML, Coq modules add no expressiveness; we can implement any module as an inhabitant of a dependent record type. It is the second-class nature of modules that makes them easier to use than dependent records in many case. Because modules may only be used in quite restricted ways, it is easier to support convenient module coding through special commands and editing modes, as the above example demonstrates. An isomorphic implementation with records would have suffered from lack of such conveniences as module subtyping and importation of the fields of a module. On the other hand, all module values must be determined statically, so modules may not be computed, e.g., within the definitions of normal functions, based on particular function parameters. *)
adamc@243 755
adamc@243 756
adamc@243 757 (** * Build Processes *)
adamc@243 758
adam@433 759 (* begin hide *)
adam@437 760 (* begin thide *)
adam@433 761 Module Lib.
adam@433 762 Module A.
adam@433 763 End A.
adam@433 764 Module B.
adam@433 765 End B.
adam@433 766 Module C.
adam@433 767 End C.
adam@433 768 End Lib.
adam@433 769 Module Client.
adam@433 770 Module D.
adam@433 771 End D.
adam@433 772 Module E.
adam@433 773 End E.
adam@433 774 End Client.
adam@437 775 (* end thide *)
adam@433 776 (* end hide *)
adam@433 777
adamc@243 778 (** As in software development, large Coq projects are much more manageable when split across multiple files and when decomposed into libraries. Coq and Proof General provide very good support for these activities.
adamc@243 779
adam@435 780 Consider a library that we will name [Lib], housed in directory <<LIB>> and split between files <<A.v>>, <<B.v>>, and <<C.v>>. A simple %\index{Makefile}%Makefile will compile the library, relying on the standard Coq tool %\index{coq\_makefile}%<<coq_makefile>> to do the hard work.
adamc@243 781
adamc@243 782 <<
adamc@243 783 MODULES := A B C
adamc@243 784 VS := $(MODULES:%=%.v)
adamc@243 785
adamc@243 786 .PHONY: coq clean
adamc@243 787
adamc@243 788 coq: Makefile.coq
adam@369 789 $(MAKE) -f Makefile.coq
adamc@243 790
adamc@243 791 Makefile.coq: Makefile $(VS)
adamc@243 792 coq_makefile -R . Lib $(VS) -o Makefile.coq
adamc@243 793
adamc@243 794 clean:: Makefile.coq
adam@369 795 $(MAKE) -f Makefile.coq clean
adamc@243 796 rm -f Makefile.coq
adamc@243 797 >>
adamc@243 798
adam@435 799 The Makefile begins by defining a variable <<VS>> holding the list of filenames to be included in the project. The primary target is <<coq>>, which depends on the construction of an auxiliary Makefile called <<Makefile.coq>>. Another rule explains how to build that file. We call <<coq_makefile>>, using the <<-R>> flag to specify that files in the current directory should be considered to belong to the library [Lib]. This Makefile will build a compiled version of each module, such that <<X.v>> is compiled into <<X.vo>>.
adamc@243 800
adam@433 801 Now code in <<B.v>> may refer to definitions in <<A.v>> after running
adamc@243 802 [[
adamc@243 803 Require Import Lib.A.
adam@367 804 ]]
adam@433 805 %\vspace{-.15in}%Library [Lib] is presented as a module, containing a submodule [A], which contains the definitions from <<A.v>>. These are genuine modules in the sense of Coq's module system, and they may be passed to functors and so on.
adamc@243 806
adam@433 807 The command [Require Import] is a convenient combination of two more primitive commands. The %\index{Vernacular commands!Require}%[Require] command finds the <<.vo>> file containing the named module, ensuring that the module is loaded into memory. The %\index{Vernacular commands!Import}%[Import] command loads all top-level definitions of the named module into the current namespace, and it may be used with local modules that do not have corresponding <<.vo>> files. Another command, %\index{Vernacular commands!Load}%[Load], is for inserting the contents of a named file verbatim. It is generally better to use the module-based commands, since they avoid rerunning proof scripts, and they facilitate reorganization of directory structure without the need to change code.
adamc@243 808
adam@433 809 Now we would like to use our library from a different development, called [Client] and found in directory <<CLIENT>>, which has its own Makefile.
adamc@243 810
adamc@243 811 <<
adamc@243 812 MODULES := D E
adamc@243 813 VS := $(MODULES:%=%.v)
adamc@243 814
adamc@243 815 .PHONY: coq clean
adamc@243 816
adamc@243 817 coq: Makefile.coq
adam@369 818 $(MAKE) -f Makefile.coq
adamc@243 819
adamc@243 820 Makefile.coq: Makefile $(VS)
adamc@243 821 coq_makefile -R LIB Lib -R . Client $(VS) -o Makefile.coq
adamc@243 822
adamc@243 823 clean:: Makefile.coq
adam@369 824 $(MAKE) -f Makefile.coq clean
adamc@243 825 rm -f Makefile.coq
adamc@243 826 >>
adamc@243 827
adam@435 828 We change the <<coq_makefile>> call to indicate where the library [Lib] is found. Now <<D.v>> and <<E.v>> can refer to definitions from [Lib] module [A] after running
adamc@243 829 [[
adamc@243 830 Require Import Lib.A.
adamc@243 831 ]]
adam@433 832 %\vspace{-.15in}\noindent{}%and <<E.v>> can refer to definitions from <<D.v>> by running
adamc@243 833 [[
adamc@243 834 Require Import Client.D.
adamc@243 835 ]]
adam@433 836 %\vspace{-.15in}%It can be useful to split a library into several files, but it is also inconvenient for client code to import library modules individually. We can get the best of both worlds by, for example, adding an extra source file <<Lib.v>> to [Lib]'s directory and Makefile, where that file contains just this line:%\index{Vernacular commands!Require Export}%
adamc@243 837 [[
adamc@243 838 Require Export Lib.A Lib.B Lib.C.
adamc@243 839 ]]
adam@367 840 %\vspace{-.15in}%Now client code can import all definitions from all of [Lib]'s modules simply by running
adamc@243 841 [[
adamc@243 842 Require Import Lib.
adamc@243 843 ]]
adam@367 844 %\vspace{-.15in}%The two Makefiles above share a lot of code, so, in practice, it is useful to define a common Makefile that is included by multiple library-specific Makefiles.
adamc@243 845
adamc@243 846 %\medskip%
adamc@243 847
adam@433 848 The remaining ingredient is the proper way of editing library code files in Proof General. Recall this snippet of <<.emacs>> code from Chapter 2, which tells Proof General where to find the library associated with this book.
adamc@243 849
adamc@243 850 <<
adamc@243 851 (custom-set-variables
adamc@243 852 ...
adamc@243 853 '(coq-prog-args '("-I" "/path/to/cpdt/src"))
adamc@243 854 ...
adamc@243 855 )
adamc@243 856 >>
adamc@243 857
adamc@243 858 To do interactive editing of our current example, we just need to change the flags to point to the right places.
adamc@243 859
adamc@243 860 <<
adamc@243 861 (custom-set-variables
adamc@243 862 ...
adamc@243 863 ; '(coq-prog-args '("-I" "/path/to/cpdt/src"))
adamc@243 864 '(coq-prog-args '("-R" "LIB" "Lib" "-R" "CLIENT" "Client"))
adamc@243 865 ...
adamc@243 866 )
adamc@243 867 >>
adamc@243 868
adam@433 869 When working on multiple projects, it is useful to leave multiple versions of this setting in your <<.emacs>> file, commenting out all but one of them at any moment in time. To switch between projects, change the commenting structure and restart Emacs.
adam@397 870
adam@433 871 Alternatively, we can revisit the directory-local settings approach and write the following into a file <<.dir-locals.el>> in <<CLIENT>>:
adam@397 872
adam@397 873 <<
adam@397 874 ((coq-mode . ((coq-prog-args .
adam@397 875 ("-emacs-U" "-R" "LIB" "Lib" "-R" "CLIENT" "Client")))))
adam@397 876 >>
adam@397 877 *)