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Pass through InductiveTypes, to incorporate new coqdoc features
author Adam Chlipala <adam@chlipala.net>
date Wed, 25 Jul 2012 16:15:50 -0400
parents 6f0f80ffd5b6
children 5f25705a10ea
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@367 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@367 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
adamc@236 52 (** We can write a very manual proof that [double] really doubles an expression's value. *)
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@387 138 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@387 159 (** 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.
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
adamc@236 197 Abort.
adamc@236 198
adamc@236 199 (** 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@236 200
adamc@236 201 Theorem eval_times : forall k e,
adamc@236 202 eval (times k e) = k * eval e.
adamc@236 203 induction e as [ | ? IHe1 ? IHe2 | ? IHe1 ? IHe2 ]; [
adamc@236 204 trivial
adamc@236 205 | simpl; rewrite IHe1; rewrite IHe2; rewrite mult_plus_distr_l; trivial
adamc@236 206 | simpl; rewrite IHe1; rewrite mult_assoc; trivial ].
adamc@236 207 Qed.
adamc@236 208
adamc@236 209 (** 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@236 210
adamc@236 211 Reset eval_times.
adamc@236 212
adam@375 213 Hint Rewrite mult_plus_distr_l.
adamc@238 214
adamc@236 215 Theorem eval_times : forall k e,
adamc@236 216 eval (times k e) = k * eval e.
adamc@236 217 induction e; crush.
adamc@236 218 Qed.
adam@368 219 (* end thide *)
adamc@236 220
adamc@237 221 (** 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@237 222
adam@398 223 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@237 224
adamc@237 225 An example should help illustrate what I mean. Consider this function, which rewrites an expression using associativity of addition and multiplication. *)
adamc@237 226
adamc@237 227 Fixpoint reassoc (e : exp) : exp :=
adamc@237 228 match e with
adamc@237 229 | Const _ => e
adamc@237 230 | Plus e1 e2 =>
adamc@237 231 let e1' := reassoc e1 in
adamc@237 232 let e2' := reassoc e2 in
adamc@237 233 match e2' with
adamc@237 234 | Plus e21 e22 => Plus (Plus e1' e21) e22
adamc@237 235 | _ => Plus e1' e2'
adamc@237 236 end
adamc@237 237 | Mult e1 e2 =>
adamc@237 238 let e1' := reassoc e1 in
adamc@237 239 let e2' := reassoc e2 in
adamc@237 240 match e2' with
adamc@237 241 | Mult e21 e22 => Mult (Mult e1' e21) e22
adamc@237 242 | _ => Mult e1' e2'
adamc@237 243 end
adamc@237 244 end.
adamc@237 245
adamc@237 246 Theorem reassoc_correct : forall e, eval (reassoc e) = eval e.
adam@368 247 (* begin thide *)
adamc@237 248 induction e; crush;
adamc@237 249 match goal with
adam@413 250 | [ |- context[match ?E with Const _ => _ | _ => _ end] ] =>
adamc@237 251 destruct E; crush
adamc@237 252 end.
adamc@237 253
adamc@237 254 (** One subgoal remains:
adamc@237 255 [[
adamc@237 256 IHe2 : eval e3 * eval e4 = eval e2
adamc@237 257 ============================
adamc@237 258 eval e1 * eval e3 * eval e4 = eval e1 * eval e2
adamc@237 259 ]]
adamc@237 260
adamc@237 261 [crush] does not know how to finish this goal. We could finish the proof manually. *)
adamc@237 262
adamc@237 263 rewrite <- IHe2; crush.
adamc@237 264
adamc@237 265 (** However, the proof would be easier to understand and maintain if we separated this insight into a separate lemma. *)
adamc@237 266
adamc@237 267 Abort.
adamc@237 268
adamc@237 269 Lemma rewr : forall a b c d, b * c = d -> a * b * c = a * d.
adamc@237 270 crush.
adamc@237 271 Qed.
adamc@237 272
adamc@237 273 Hint Resolve rewr.
adamc@237 274
adamc@237 275 Theorem reassoc_correct : forall e, eval (reassoc e) = eval e.
adamc@237 276 induction e; crush;
adamc@237 277 match goal with
adam@413 278 | [ |- context[match ?E with Const _ => _ | _ => _ end] ] =>
adamc@237 279 destruct E; crush
adamc@237 280 end.
adamc@237 281 Qed.
adam@368 282 (* end thide *)
adamc@237 283
adamc@237 284 (** 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@237 285
adam@387 286 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@387 287
adam@413 288 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@387 289
adam@387 290 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@387 291
adam@398 292 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@237 293
adamc@235 294
adamc@238 295 (** * Debugging and Maintaining Automation *)
adamc@238 296
adam@367 297 (** 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@238 298
adam@367 299 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@238 300
adam@387 301 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@238 302
adamc@238 303 (* begin hide *)
adamc@238 304 Require Import MoreDep.
adamc@238 305 (* end hide *)
adamc@238 306
adamc@238 307 Theorem cfold_correct : forall t (e : exp t), expDenote e = expDenote (cfold e).
adam@368 308 (* begin thide *)
adamc@238 309 induction e; crush.
adamc@238 310
adamc@238 311 dep_destruct (cfold e1); crush.
adamc@238 312 dep_destruct (cfold e2); crush.
adamc@238 313
adamc@238 314 dep_destruct (cfold e1); crush.
adamc@238 315 dep_destruct (cfold e2); crush.
adamc@238 316
adamc@238 317 dep_destruct (cfold e1); crush.
adamc@238 318 dep_destruct (cfold e2); crush.
adamc@238 319
adamc@238 320 dep_destruct (cfold e1); crush.
adamc@238 321 dep_destruct (expDenote e1); crush.
adamc@238 322
adamc@238 323 dep_destruct (cfold e); crush.
adamc@238 324
adamc@238 325 dep_destruct (cfold e); crush.
adamc@238 326 Qed.
adamc@238 327
adamc@238 328 (** In this complete proof, it is hard to avoid noticing a pattern. We rework the proof, abstracting over the patterns we find. *)
adamc@238 329
adamc@238 330 Reset cfold_correct.
adamc@238 331
adamc@238 332 Theorem cfold_correct : forall t (e : exp t), expDenote e = expDenote (cfold e).
adamc@238 333 induction e; crush.
adamc@238 334
adamc@238 335 (** 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@238 336
adamc@238 337 Ltac t :=
adamc@238 338 repeat (match goal with
adam@413 339 | [ |- context[match ?E with NConst _ => _ | _ => _ end] ] =>
adamc@238 340 dep_destruct E
adamc@238 341 end; crush).
adamc@238 342
adamc@238 343 t.
adamc@238 344
adamc@238 345 (** 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@238 346
adamc@238 347 t.
adamc@238 348
adamc@238 349 t.
adamc@238 350
adamc@238 351 t.
adamc@238 352
adamc@238 353 (** The subgoal's conclusion is:
adamc@238 354 [[
adamc@238 355 ============================
adamc@238 356 (if expDenote e1 then expDenote (cfold e2) else expDenote (cfold e3)) =
adamc@238 357 expDenote (if expDenote e1 then cfold e2 else cfold e3)
adamc@238 358 ]]
adamc@238 359
adamc@238 360 We need to expand our [t] tactic to handle this case. *)
adamc@238 361
adamc@238 362 Ltac t' :=
adamc@238 363 repeat (match goal with
adam@413 364 | [ |- context[match ?E with NConst _ => _ | _ => _ end] ] =>
adamc@238 365 dep_destruct E
adamc@238 366 | [ |- (if ?E then _ else _) = _ ] => destruct E
adamc@238 367 end; crush).
adamc@238 368
adamc@238 369 t'.
adamc@238 370
adamc@238 371 (** Now the goal is discharged, but [t'] has no effect on the next subgoal. *)
adamc@238 372
adamc@238 373 t'.
adamc@238 374
adamc@238 375 (** A final revision of [t] finishes the proof. *)
adamc@238 376
adamc@238 377 Ltac t'' :=
adamc@238 378 repeat (match goal with
adam@413 379 | [ |- context[match ?E with NConst _ => _ | _ => _ end] ] =>
adamc@238 380 dep_destruct E
adamc@238 381 | [ |- (if ?E then _ else _) = _ ] => destruct E
adamc@238 382 | [ |- context[match pairOut ?E with Some _ => _
adamc@238 383 | None => _ end] ] =>
adamc@238 384 dep_destruct E
adamc@238 385 end; crush).
adamc@238 386
adamc@238 387 t''.
adamc@238 388
adamc@238 389 t''.
adamc@238 390 Qed.
adamc@238 391
adam@367 392 (** We can take the final tactic and move it into the initial part of the proof script, arriving at a nicely automated proof. *)
adamc@238 393
adamc@238 394 Reset t.
adamc@238 395
adamc@238 396 Theorem cfold_correct : forall t (e : exp t), expDenote e = expDenote (cfold e).
adamc@238 397 induction e; crush;
adamc@238 398 repeat (match goal with
adam@413 399 | [ |- context[match ?E with NConst _ => _ | _ => _ end] ] =>
adamc@238 400 dep_destruct E
adamc@238 401 | [ |- (if ?E then _ else _) = _ ] => destruct E
adamc@238 402 | [ |- context[match pairOut ?E with Some _ => _
adamc@238 403 | None => _ end] ] =>
adamc@238 404 dep_destruct E
adamc@238 405 end; crush).
adamc@238 406 Qed.
adam@368 407 (* end thide *)
adamc@238 408
adam@367 409 (** 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@240 410
adamc@240 411 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@240 412
adamc@240 413 Reset reassoc_correct.
adamc@240 414
adamc@240 415 Theorem confounder : forall e1 e2 e3,
adamc@240 416 eval e1 * eval e2 * eval e3 = eval e1 * (eval e2 + 1 - 1) * eval e3.
adamc@240 417 crush.
adamc@240 418 Qed.
adamc@240 419
adam@375 420 Hint Rewrite confounder.
adamc@240 421
adamc@240 422 Theorem reassoc_correct : forall e, eval (reassoc e) = eval e.
adam@368 423 (* begin thide *)
adamc@240 424 induction e; crush;
adamc@240 425 match goal with
adam@413 426 | [ |- context[match ?E with Const _ => _ | _ => _ end] ] =>
adamc@240 427 destruct E; crush
adamc@240 428 end.
adamc@240 429
adamc@240 430 (** One subgoal remains:
adamc@240 431
adamc@240 432 [[
adamc@240 433 ============================
adamc@240 434 eval e1 * (eval e3 + 1 - 1) * eval e4 = eval e1 * eval e2
adamc@240 435 ]]
adamc@240 436
adam@367 437 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@240 438
adamc@240 439 Restart.
adamc@240 440
adamc@240 441 Ltac t := crush; match goal with
adam@413 442 | [ |- context[match ?E with Const _ => _ | _ => _ end] ] =>
adamc@240 443 destruct E; crush
adamc@240 444 end.
adamc@240 445
adamc@240 446 induction e.
adamc@240 447
adam@387 448 (** 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@240 449
adamc@240 450 t.
adamc@240 451
adamc@240 452 (** The next subgoal, for addition, is also discharged without trouble. *)
adamc@240 453
adamc@240 454 t.
adamc@240 455
adamc@240 456 (** The final subgoal is for multiplication, and it is here that we get stuck in the proof state summarized above. *)
adamc@240 457
adamc@240 458 t.
adamc@240 459
adam@367 460 (** What is [t] doing to get us to this point? The %\index{tactics!info}%[info] command can help us answer this kind of question. *)
adamc@240 461
adamc@240 462 Undo.
adamc@240 463 info t.
adam@413 464
adam@367 465 (** %\vspace{-.15in}%[[
adam@375 466 == simpl in *; intuition; subst; autorewrite with core in *;
adam@375 467 simpl in *; intuition; subst; autorewrite with core in *;
adamc@240 468 simpl in *; intuition; subst; destruct (reassoc e2).
adamc@240 469 simpl in *; intuition.
adamc@240 470
adamc@240 471 simpl in *; intuition.
adamc@240 472
adam@375 473 simpl in *; intuition; subst; autorewrite with core in *;
adamc@240 474 refine (eq_ind_r
adamc@240 475 (fun n : nat =>
adamc@240 476 n * (eval e3 + 1 - 1) * eval e4 = eval e1 * eval e2) _ IHe1);
adam@375 477 autorewrite with core in *; simpl in *; intuition;
adam@375 478 subst; autorewrite with core in *; simpl in *;
adamc@240 479 intuition; subst.
adamc@240 480
adamc@240 481 ]]
adamc@240 482
adamc@240 483 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@240 484
adamc@240 485 Undo.
adamc@240 486
adamc@240 487 (** We arbitrarily split the script into chunks. The first few seem not to do any harm. *)
adamc@240 488
adam@375 489 simpl in *; intuition; subst; autorewrite with core in *.
adam@375 490 simpl in *; intuition; subst; autorewrite with core in *.
adamc@240 491 simpl in *; intuition; subst; destruct (reassoc e2).
adamc@240 492 simpl in *; intuition.
adamc@240 493 simpl in *; intuition.
adamc@240 494
adamc@240 495 (** The next step is revealed as the culprit, bringing us to the final unproved subgoal. *)
adamc@240 496
adam@375 497 simpl in *; intuition; subst; autorewrite with core in *.
adamc@240 498
adamc@240 499 (** We can split the steps further to assign blame. *)
adamc@240 500
adamc@240 501 Undo.
adamc@240 502
adamc@240 503 simpl in *.
adamc@240 504 intuition.
adamc@240 505 subst.
adam@375 506 autorewrite with core in *.
adamc@240 507
adamc@240 508 (** 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@240 509
adamc@240 510 Undo.
adamc@240 511
adam@375 512 info autorewrite with core in *.
adam@367 513 (** %\vspace{-.15in}%[[
adamc@240 514 == refine (eq_ind_r (fun n : nat => n = eval e1 * eval e2) _
adamc@240 515 (confounder (reassoc e1) e3 e4)).
adamc@240 516 ]]
adamc@240 517
adamc@240 518 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@240 519
adamc@240 520 Abort.
adam@368 521 (* end thide *)
adamc@240 522
adamc@241 523 (** 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@241 524
adamc@241 525 Here is one example of a performance surprise. *)
adamc@241 526
adamc@239 527 Section slow.
adamc@239 528 Hint Resolve trans_eq.
adamc@239 529
adamc@241 530 (** 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@241 531
adamc@239 532 Variable A : Set.
adamc@239 533 Variables P Q R S : A -> A -> Prop.
adamc@239 534 Variable f : A -> A.
adamc@239 535
adamc@239 536 Hypothesis H1 : forall x y, P x y -> Q x y -> R x y -> f x = f y.
adamc@239 537 Hypothesis H2 : forall x y, S x y -> R x y.
adamc@239 538
adam@367 539 (** We prove a simple lemma very quickly, using the %\index{Vernacular commands!Time}%[Time] command to measure exactly how quickly. *)
adamc@241 540
adamc@239 541 Lemma slow : forall x y, P x y -> Q x y -> S x y -> f x = f y.
adamc@241 542 Time eauto 6.
adam@367 543 (** %\vspace{-.2in}%[[
adamc@241 544 Finished transaction in 0. secs (0.068004u,0.s)
adam@302 545 ]]
adam@302 546 *)
adamc@241 547
adamc@239 548 Qed.
adamc@239 549
adamc@241 550 (** Now we add a different hypothesis, which is innocent enough; in fact, it is even provable as a theorem. *)
adamc@241 551
adamc@239 552 Hypothesis H3 : forall x y, x = y -> f x = f y.
adamc@239 553
adamc@239 554 Lemma slow' : forall x y, P x y -> Q x y -> S x y -> f x = f y.
adamc@241 555 Time eauto 6.
adam@367 556 (** %\vspace{-.2in}%[[
adamc@241 557 Finished transaction in 2. secs (1.264079u,0.s)
adamc@241 558 ]]
adamc@241 559
adamc@241 560 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@241 561
adam@368 562 (* begin thide *)
adamc@241 563 Restart.
adamc@241 564 info eauto 6.
adam@367 565 (** %\vspace{-.15in}%[[
adamc@241 566 == intro x; intro y; intro H; intro H0; intro H4;
adamc@241 567 simple eapply trans_eq.
adamc@241 568 simple apply refl_equal.
adamc@241 569
adamc@241 570 simple eapply trans_eq.
adamc@241 571 simple apply refl_equal.
adamc@241 572
adamc@241 573 simple eapply trans_eq.
adamc@241 574 simple apply refl_equal.
adamc@241 575
adamc@241 576 simple apply H1.
adamc@241 577 eexact H.
adamc@241 578
adamc@241 579 eexact H0.
adamc@241 580
adamc@241 581 simple apply H2; eexact H4.
adamc@241 582 ]]
adamc@241 583
adam@367 584 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@241 585
adamc@241 586 Restart.
adamc@241 587 debug eauto 6.
adamc@241 588
adamc@241 589 (** The output is a large proof tree. The beginning of the tree is enough to reveal what is happening:
adamc@241 590 [[
adamc@241 591 1 depth=6
adamc@241 592 1.1 depth=6 intro
adamc@241 593 1.1.1 depth=6 intro
adamc@241 594 1.1.1.1 depth=6 intro
adamc@241 595 1.1.1.1.1 depth=6 intro
adamc@241 596 1.1.1.1.1.1 depth=6 intro
adamc@241 597 1.1.1.1.1.1.1 depth=5 apply H3
adamc@241 598 1.1.1.1.1.1.1.1 depth=4 eapply trans_eq
adamc@241 599 1.1.1.1.1.1.1.1.1 depth=4 apply refl_equal
adamc@241 600 1.1.1.1.1.1.1.1.1.1 depth=3 eapply trans_eq
adamc@241 601 1.1.1.1.1.1.1.1.1.1.1 depth=3 apply refl_equal
adamc@241 602 1.1.1.1.1.1.1.1.1.1.1.1 depth=2 eapply trans_eq
adamc@241 603 1.1.1.1.1.1.1.1.1.1.1.1.1 depth=2 apply refl_equal
adamc@241 604 1.1.1.1.1.1.1.1.1.1.1.1.1.1 depth=1 eapply trans_eq
adamc@241 605 1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 depth=1 apply refl_equal
adamc@241 606 1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 depth=0 eapply trans_eq
adamc@241 607 1.1.1.1.1.1.1.1.1.1.1.1.1.1.2 depth=1 apply sym_eq ; trivial
adamc@241 608 1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1 depth=0 eapply trans_eq
adamc@241 609 1.1.1.1.1.1.1.1.1.1.1.1.1.1.3 depth=0 eapply trans_eq
adamc@241 610 1.1.1.1.1.1.1.1.1.1.1.1.2 depth=2 apply sym_eq ; trivial
adamc@241 611 1.1.1.1.1.1.1.1.1.1.1.1.2.1 depth=1 eapply trans_eq
adamc@241 612 1.1.1.1.1.1.1.1.1.1.1.1.2.1.1 depth=1 apply refl_equal
adamc@241 613 1.1.1.1.1.1.1.1.1.1.1.1.2.1.1.1 depth=0 eapply trans_eq
adamc@241 614 1.1.1.1.1.1.1.1.1.1.1.1.2.1.2 depth=1 apply sym_eq ; trivial
adamc@241 615 1.1.1.1.1.1.1.1.1.1.1.1.2.1.2.1 depth=0 eapply trans_eq
adamc@241 616 1.1.1.1.1.1.1.1.1.1.1.1.2.1.3 depth=0 eapply trans_eq
adamc@241 617 ]]
adamc@241 618
adam@367 619 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@241 620
adamc@239 621 Qed.
adam@368 622 (* end thide *)
adamc@239 623 End slow.
adamc@239 624
adam@387 625 (** 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@387 626
adam@413 627 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@241 628
adam@367 629 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 remaining. Still, many large automated proofs can realize vast memory savings via [abstract]. *)
adamc@241 630
adamc@238 631
adamc@235 632 (** * Modules *)
adamc@235 633
adam@398 634 (** 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 Objective Caml, and the discussion that follows assumes familiarity with the basics of one of those systems.
adamc@242 635
adam@413 636 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@242 637
adam@367 638 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, this 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 [e] for [f], and an operation [i] that is a left inverse for [f].%\index{Vernacular commands!Module Type}% *)
adamc@242 639
adamc@235 640 Module Type GROUP.
adamc@235 641 Parameter G : Set.
adamc@235 642 Parameter f : G -> G -> G.
adamc@235 643 Parameter e : G.
adamc@235 644 Parameter i : G -> G.
adamc@235 645
adamc@235 646 Axiom assoc : forall a b c, f (f a b) c = f a (f b c).
adamc@235 647 Axiom ident : forall a, f e a = a.
adamc@235 648 Axiom inverse : forall a, f (i a) a = e.
adamc@235 649 End GROUP.
adamc@235 650
adam@367 651 (** Many useful theorems hold of arbitrary groups. We capture some such theorem statements in another module signature.%\index{Vernacular commands!Declare Module}% *)
adamc@242 652
adamc@235 653 Module Type GROUP_THEOREMS.
adamc@235 654 Declare Module M : GROUP.
adamc@235 655
adamc@235 656 Axiom ident' : forall a, M.f a M.e = a.
adamc@235 657
adamc@235 658 Axiom inverse' : forall a, M.f a (M.i a) = M.e.
adamc@235 659
adamc@235 660 Axiom unique_ident : forall e', (forall a, M.f e' a = a) -> e' = M.e.
adamc@235 661 End GROUP_THEOREMS.
adamc@235 662
adam@387 663 (** We implement generic proofs of these theorems with a functor, whose input is an arbitrary group [M]. %\index{Vernacular commands!Module}% *)
adamc@242 664
adam@387 665 Module GroupProofs (M : GROUP) : GROUP_THEOREMS with Module M := M.
adam@413 666 (** 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@387 667
adamc@235 668 Module M := M.
adam@387 669 (** 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@235 670
adamc@235 671 Import M.
adam@387 672 (** 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@387 673
adam@387 674 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@235 675
adamc@235 676 Theorem inverse' : forall a, f a (i a) = e.
adamc@235 677 intro.
adamc@235 678 rewrite <- (ident (f a (i a))).
adamc@235 679 rewrite <- (inverse (f a (i a))) at 1.
adamc@235 680 rewrite assoc.
adamc@235 681 rewrite assoc.
adamc@235 682 rewrite <- (assoc (i a) a (i a)).
adamc@235 683 rewrite inverse.
adamc@235 684 rewrite ident.
adamc@235 685 apply inverse.
adamc@235 686 Qed.
adamc@235 687
adamc@235 688 Theorem ident' : forall a, f a e = a.
adamc@235 689 intro.
adamc@235 690 rewrite <- (inverse a).
adamc@235 691 rewrite <- assoc.
adamc@235 692 rewrite inverse'.
adamc@235 693 apply ident.
adamc@235 694 Qed.
adamc@235 695
adamc@235 696 Theorem unique_ident : forall e', (forall a, M.f e' a = a) -> e' = M.e.
adamc@235 697 intros.
adamc@235 698 rewrite <- (H e).
adamc@235 699 symmetry.
adamc@235 700 apply ident'.
adamc@235 701 Qed.
adam@387 702 End GroupProofs.
adamc@239 703
adamc@242 704 (** We can show that the integers with [+] form a group. *)
adamc@242 705
adamc@239 706 Require Import ZArith.
adamc@239 707 Open Scope Z_scope.
adamc@239 708
adamc@239 709 Module Int.
adamc@239 710 Definition G := Z.
adamc@239 711 Definition f x y := x + y.
adamc@239 712 Definition e := 0.
adamc@239 713 Definition i x := -x.
adamc@239 714
adamc@239 715 Theorem assoc : forall a b c, f (f a b) c = f a (f b c).
adamc@239 716 unfold f; crush.
adamc@239 717 Qed.
adamc@239 718 Theorem ident : forall a, f e a = a.
adamc@239 719 unfold f, e; crush.
adamc@239 720 Qed.
adamc@239 721 Theorem inverse : forall a, f (i a) a = e.
adamc@239 722 unfold f, i, e; crush.
adamc@239 723 Qed.
adamc@239 724 End Int.
adamc@239 725
adamc@242 726 (** Next, we can produce integer-specific versions of the generic group theorems. *)
adamc@242 727
adam@387 728 Module IntProofs := GroupProofs(Int).
adamc@239 729
adam@387 730 Check IntProofs.unique_ident.
adamc@242 731 (** %\vspace{-.15in}% [[
adam@387 732 IntProofs.unique_ident
adamc@242 733 : forall e' : Int.G, (forall a : Int.G, Int.f e' a = a) -> e' = Int.e
adam@302 734 ]]
adam@367 735
adam@367 736 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@239 737
adamc@239 738 Theorem unique_ident : forall e', (forall a, e' + a = a) -> e' = 0.
adam@368 739 (* begin thide *)
adam@387 740 exact IntProofs.unique_ident.
adamc@239 741 Qed.
adam@368 742 (* end thide *)
adamc@242 743
adam@367 744 (** 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 defintions of normal functions, based on particular function parameters. *)
adamc@243 745
adamc@243 746
adamc@243 747 (** * Build Processes *)
adamc@243 748
adamc@243 749 (** 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@243 750
adam@367 751 Consider a library that we will name [Lib], housed in directory %\texttt{%#<tt>#LIB#</tt>#%}% and split between files %\texttt{%#<tt>#A.v#</tt>#%}%, %\texttt{%#<tt>#B.v#</tt>#%}%, and %\texttt{%#<tt>#C.v#</tt>#%}%. A simple %\index{Makefile}%Makefile will compile the library, relying on the standard Coq tool %\index{coq\_makefile}\texttt{%#<tt>#coq_makefile#</tt>#%}% to do the hard work.
adamc@243 752
adamc@243 753 <<
adamc@243 754 MODULES := A B C
adamc@243 755 VS := $(MODULES:%=%.v)
adamc@243 756
adamc@243 757 .PHONY: coq clean
adamc@243 758
adamc@243 759 coq: Makefile.coq
adam@369 760 $(MAKE) -f Makefile.coq
adamc@243 761
adamc@243 762 Makefile.coq: Makefile $(VS)
adamc@243 763 coq_makefile -R . Lib $(VS) -o Makefile.coq
adamc@243 764
adamc@243 765 clean:: Makefile.coq
adam@369 766 $(MAKE) -f Makefile.coq clean
adamc@243 767 rm -f Makefile.coq
adamc@243 768 >>
adamc@243 769
adamc@243 770 The Makefile begins by defining a variable %\texttt{%#<tt>#VS#</tt>#%}% holding the list of filenames to be included in the project. The primary target is %\texttt{%#<tt>#coq#</tt>#%}%, which depends on the construction of an auxiliary Makefile called %\texttt{%#<tt>#Makefile.coq#</tt>#%}%. Another rule explains how to build that file. We call %\texttt{%#<tt>#coq_makefile#</tt>#%}%, using the %\texttt{%#<tt>#-R#</tt>#%}% 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 %\texttt{%#<tt>#X.v#</tt>#%}% is compiled into %\texttt{%#<tt>#X.vo#</tt>#%}%.
adamc@243 771
adamc@243 772 Now code in %\texttt{%#<tt>#B.v#</tt>#%}% may refer to definitions in %\texttt{%#<tt>#A.v#</tt>#%}% after running
adamc@243 773 [[
adamc@243 774 Require Import Lib.A.
adam@367 775 ]]
adam@367 776 %\vspace{-.15in}%Library [Lib] is presented as a module, containing a submodule [A], which contains the definitions from %\texttt{%#<tt>#A.v#</tt>#%}%. These are genuine modules in the sense of Coq's module system, and they may be passed to functors and so on.
adamc@243 777
adam@367 778 The command [Require Import] is a convenient combination of two more primitive commands. The %\index{Vernacular commands!Require}%[Require] command finds the %\texttt{%#<tt>#.vo#</tt>#%}% 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 %\texttt{%#<tt>#.vo#</tt>#%}% 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@243 779
adamc@243 780 Now we would like to use our library from a different development, called [Client] and found in directory %\texttt{%#<tt>#CLIENT#</tt>#%}%, which has its own Makefile.
adamc@243 781
adamc@243 782 <<
adamc@243 783 MODULES := D E
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 Lib -R . Client $(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
adamc@243 799 We change the %\texttt{%#<tt>#coq_makefile#</tt>#%}% call to indicate where the library [Lib] is found. Now %\texttt{%#<tt>#D.v#</tt>#%}% and %\texttt{%#<tt>#E.v#</tt>#%}% can refer to definitions from [Lib] module [A] after running
adamc@243 800 [[
adamc@243 801 Require Import Lib.A.
adamc@243 802 ]]
adam@367 803 %\vspace{-.15in}\noindent{}%and %\texttt{%#<tt>#E.v#</tt>#%}% can refer to definitions from %\texttt{%#<tt>#D.v#</tt>#%}% by running
adamc@243 804 [[
adamc@243 805 Require Import Client.D.
adamc@243 806 ]]
adam@367 807 %\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 %\texttt{%#<tt>#Lib.v#</tt>#%}% to [Lib]'s directory and Makefile, where that file contains just this line:%\index{Vernacular commands!Require Export}%
adamc@243 808 [[
adamc@243 809 Require Export Lib.A Lib.B Lib.C.
adamc@243 810 ]]
adam@367 811 %\vspace{-.15in}%Now client code can import all definitions from all of [Lib]'s modules simply by running
adamc@243 812 [[
adamc@243 813 Require Import Lib.
adamc@243 814 ]]
adam@367 815 %\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@243 816
adamc@243 817 %\medskip%
adamc@243 818
adamc@243 819 The remaining ingredient is the proper way of editing library code files in Proof General. Recall this snippet of %\texttt{%#<tt>#.emacs#</tt>#%}% code from Chapter 2, which tells Proof General where to find the library associated with this book.
adamc@243 820
adamc@243 821 <<
adamc@243 822 (custom-set-variables
adamc@243 823 ...
adamc@243 824 '(coq-prog-args '("-I" "/path/to/cpdt/src"))
adamc@243 825 ...
adamc@243 826 )
adamc@243 827 >>
adamc@243 828
adamc@243 829 To do interactive editing of our current example, we just need to change the flags to point to the right places.
adamc@243 830
adamc@243 831 <<
adamc@243 832 (custom-set-variables
adamc@243 833 ...
adamc@243 834 ; '(coq-prog-args '("-I" "/path/to/cpdt/src"))
adamc@243 835 '(coq-prog-args '("-R" "LIB" "Lib" "-R" "CLIENT" "Client"))
adamc@243 836 ...
adamc@243 837 )
adamc@243 838 >>
adamc@243 839
adam@397 840 When working on multiple projects, it is useful to leave multiple versions of this setting in your %\texttt{%#<tt>#.emacs#</tt>#%}% 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@397 841
adam@398 842 Alternatively, we can revisit the directory-local settings approach and write the following into a file %\texttt{%#<tt>#.dir-locals.el#</tt>#%}% in %\texttt{%#<tt>#CLIENT#</tt>#%}%:
adam@397 843
adam@397 844 <<
adam@397 845 ((coq-mode . ((coq-prog-args .
adam@397 846 ("-emacs-U" "-R" "LIB" "Lib" "-R" "CLIENT" "Client")))))
adam@397 847 >>
adam@397 848 *)