## Mercurial > cpdt > repo

### annotate src/Large.v @ 535:dac7a2705b00

Find changesets by keywords (author, files, the commit message), revision
number or hash, or revset expression.

...and back to working in 8.4 again

author | Adam Chlipala <adam@chlipala.net> |
---|---|

date | Wed, 05 Aug 2015 14:57:14 -0400 |

parents | ed829eaa91b2 |

children | f874c163f5e0 |

rev | line source |
---|---|

adam@534 | 1 (* Copyright (c) 2009-2012, 2015, 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@534 | 13 Require Import Cpdt.CpdtTactics. |

adamc@235 | 14 |

adamc@235 | 15 Set Implicit Arguments. |

adam@534 | 16 Set Asymmetric Patterns. |

adamc@235 | 17 (* end hide *) |

adamc@235 | 18 |

adamc@235 | 19 |

adam@381 | 20 (** %\part{The Big Picture} |

adam@381 | 21 |

adam@381 | 22 \chapter{Proving in the Large}% *) |

adamc@235 | 23 |

adam@433 | 24 (** 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 | 25 |

adamc@236 | 26 Thus, this chapter gives some tips for structuring and maintaining large Coq developments. *) |

adamc@236 | 27 |

adamc@236 | 28 |

adamc@236 | 29 (** * Ltac Anti-Patterns *) |

adamc@236 | 30 |

adam@433 | 31 (** 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 | 32 |

adamc@237 | 33 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 | 34 |

adamc@236 | 35 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 | 36 |

adamc@236 | 37 Inductive exp : Set := |

adamc@236 | 38 | Const : nat -> exp |

adamc@236 | 39 | Plus : exp -> exp -> exp. |

adamc@236 | 40 |

adamc@236 | 41 Fixpoint eval (e : exp) : nat := |

adamc@236 | 42 match e with |

adamc@236 | 43 | Const n => n |

adamc@236 | 44 | Plus e1 e2 => eval e1 + eval e2 |

adamc@236 | 45 end. |

adamc@236 | 46 |

adamc@236 | 47 Fixpoint times (k : nat) (e : exp) : exp := |

adamc@236 | 48 match e with |

adamc@236 | 49 | Const n => Const (k * n) |

adamc@236 | 50 | Plus e1 e2 => Plus (times k e1) (times k e2) |

adamc@236 | 51 end. |

adamc@236 | 52 |

adam@452 | 53 (** We can write a very manual proof that [times] really implements multiplication. *) |

adamc@236 | 54 |

adamc@236 | 55 Theorem eval_times : forall k e, |

adamc@236 | 56 eval (times k e) = k * eval e. |

adamc@236 | 57 induction e. |

adamc@236 | 58 |

adamc@236 | 59 trivial. |

adamc@236 | 60 |

adamc@236 | 61 simpl. |

adamc@236 | 62 rewrite IHe1. |

adamc@236 | 63 rewrite IHe2. |

adamc@236 | 64 rewrite mult_plus_distr_l. |

adamc@236 | 65 trivial. |

adamc@236 | 66 Qed. |

adamc@236 | 67 |

adam@368 | 68 (* begin thide *) |

adam@367 | 69 (** 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 | 70 |

adamc@236 | 71 Reset eval_times. |

adamc@236 | 72 |

adam@368 | 73 Theorem eval_times : forall k x, |

adamc@236 | 74 eval (times k x) = k * eval x. |

adamc@236 | 75 induction x. |

adamc@236 | 76 |

adamc@236 | 77 trivial. |

adamc@236 | 78 |

adamc@236 | 79 simpl. |

adam@367 | 80 (** %\vspace{-.15in}%[[ |

adamc@236 | 81 rewrite IHe1. |

adam@367 | 82 ]] |

adamc@236 | 83 |

adam@367 | 84 << |

adamc@236 | 85 Error: The reference IHe1 was not found in the current environment. |

adam@367 | 86 >> |

adamc@236 | 87 |

adamc@236 | 88 The inductive hypotheses are named [IHx1] and [IHx2] now, not [IHe1] and [IHe2]. *) |

adamc@236 | 89 |

adamc@236 | 90 Abort. |

adamc@236 | 91 |

adamc@236 | 92 (** 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 | 93 |

adamc@236 | 94 Theorem eval_times : forall k e, |

adamc@236 | 95 eval (times k e) = k * eval e. |

adamc@236 | 96 induction e as [ | ? IHe1 ? IHe2 ]. |

adamc@236 | 97 |

adamc@236 | 98 trivial. |

adamc@236 | 99 |

adamc@236 | 100 simpl. |

adamc@236 | 101 rewrite IHe1. |

adamc@236 | 102 rewrite IHe2. |

adamc@236 | 103 rewrite mult_plus_distr_l. |

adamc@236 | 104 trivial. |

adamc@236 | 105 Qed. |

adamc@236 | 106 |

adam@509 | 107 (** We pass %\index{tactics!induction}%[induction] an%\index{intro pattern}% _intro pattern_, using a [|] character to separate 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 | 108 |

adamc@237 | 109 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 | 110 |

adamc@236 | 111 Reset times. |

adamc@236 | 112 |

adamc@236 | 113 Fixpoint times (k : nat) (e : exp) : exp := |

adamc@236 | 114 match e with |

adamc@236 | 115 | Const n => Const (1 + k * n) |

adamc@236 | 116 | Plus e1 e2 => Plus (times k e1) (times k e2) |

adamc@236 | 117 end. |

adamc@236 | 118 |

adamc@236 | 119 Theorem eval_times : forall k e, |

adamc@236 | 120 eval (times k e) = k * eval e. |

adamc@236 | 121 induction e as [ | ? IHe1 ? IHe2 ]. |

adamc@236 | 122 |

adamc@236 | 123 trivial. |

adamc@236 | 124 |

adamc@236 | 125 simpl. |

adam@367 | 126 (** %\vspace{-.15in}%[[ |

adamc@236 | 127 rewrite IHe1. |

adam@367 | 128 ]] |

adamc@236 | 129 |

adam@367 | 130 << |

adamc@236 | 131 Error: The reference IHe1 was not found in the current environment. |

adam@367 | 132 >> |

adam@302 | 133 *) |

adamc@236 | 134 |

adamc@236 | 135 Abort. |

adamc@236 | 136 |

adam@398 | 137 (** 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 | 138 |

adam@433 | 139 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 | 140 |

adamc@236 | 141 Reset times. |

adamc@236 | 142 |

adamc@236 | 143 Fixpoint times (k : nat) (e : exp) : exp := |

adamc@236 | 144 match e with |

adamc@236 | 145 | Const n => Const (k * n) |

adamc@236 | 146 | Plus e1 e2 => Plus (times k e1) (times k e2) |

adamc@236 | 147 end. |

adamc@236 | 148 |

adam@387 | 149 (** 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 | 150 |

adamc@236 | 151 We can rewrite the current proof with a single tactic. *) |

adamc@236 | 152 |

adamc@236 | 153 Theorem eval_times : forall k e, |

adamc@236 | 154 eval (times k e) = k * eval e. |

adamc@236 | 155 induction e as [ | ? IHe1 ? IHe2 ]; [ |

adamc@236 | 156 trivial |

adamc@236 | 157 | simpl; rewrite IHe1; rewrite IHe2; rewrite mult_plus_distr_l; trivial ]. |

adamc@236 | 158 Qed. |

adamc@236 | 159 |

adam@509 | 160 (** We use the form of the semicolon operator that allows a different tactic to be specified for each generated subgoal. This change improves the 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 using 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 | 161 |

adamc@236 | 162 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 | 163 |

adamc@236 | 164 Reset exp. |

adamc@236 | 165 |

adamc@236 | 166 Inductive exp : Set := |

adamc@236 | 167 | Const : nat -> exp |

adamc@236 | 168 | Plus : exp -> exp -> exp |

adamc@236 | 169 | Mult : exp -> exp -> exp. |

adamc@236 | 170 |

adamc@236 | 171 Fixpoint eval (e : exp) : nat := |

adamc@236 | 172 match e with |

adamc@236 | 173 | Const n => n |

adamc@236 | 174 | Plus e1 e2 => eval e1 + eval e2 |

adamc@236 | 175 | Mult e1 e2 => eval e1 * eval e2 |

adamc@236 | 176 end. |

adamc@236 | 177 |

adamc@236 | 178 Fixpoint times (k : nat) (e : exp) : exp := |

adamc@236 | 179 match e with |

adamc@236 | 180 | Const n => Const (k * n) |

adamc@236 | 181 | Plus e1 e2 => Plus (times k e1) (times k e2) |

adamc@236 | 182 | Mult e1 e2 => Mult (times k e1) e2 |

adamc@236 | 183 end. |

adamc@236 | 184 |

adamc@236 | 185 Theorem eval_times : forall k e, |

adamc@236 | 186 eval (times k e) = k * eval e. |

adam@367 | 187 (** %\vspace{-.25in}%[[ |

adamc@236 | 188 induction e as [ | ? IHe1 ? IHe2 ]; [ |

adamc@236 | 189 trivial |

adamc@236 | 190 | simpl; rewrite IHe1; rewrite IHe2; rewrite mult_plus_distr_l; trivial ]. |

adam@367 | 191 ]] |

adamc@236 | 192 |

adam@367 | 193 << |

adamc@236 | 194 Error: Expects a disjunctive pattern with 3 branches. |

adam@367 | 195 >> |

adam@302 | 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 |

adam@491 | 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 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 |

adam@433 | 261 The [crush] tactic 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@509 | 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, which is just what motivates the techniques outlined above, 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 |

adam@534 | 394 Reset cfold_correct. |

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@509 | 448 (** Since we see the subgoals before any simplification occurs, it is clear that we are looking at 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@433 | 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. (As of this writing, [info] is no longer functioning in the most recent Coq release, but I hope it returns.) *) |

adamc@240 | 461 |

adamc@240 | 462 Undo. |

adamc@240 | 463 info t. |

adam@413 | 464 |

adam@433 | 465 (* begin hide *) |

adam@437 | 466 (* begin thide *) |

adam@433 | 467 Definition eir := eq_ind_r. |

adam@437 | 468 (* end thide *) |

adam@433 | 469 (* end hide *) |

adam@433 | 470 |

adam@367 | 471 (** %\vspace{-.15in}%[[ |

adam@375 | 472 == simpl in *; intuition; subst; autorewrite with core in *; |

adam@375 | 473 simpl in *; intuition; subst; autorewrite with core in *; |

adamc@240 | 474 simpl in *; intuition; subst; destruct (reassoc e2). |

adamc@240 | 475 simpl in *; intuition. |

adamc@240 | 476 |

adamc@240 | 477 simpl in *; intuition. |

adamc@240 | 478 |

adam@375 | 479 simpl in *; intuition; subst; autorewrite with core in *; |

adamc@240 | 480 refine (eq_ind_r |

adamc@240 | 481 (fun n : nat => |

adamc@240 | 482 n * (eval e3 + 1 - 1) * eval e4 = eval e1 * eval e2) _ IHe1); |

adam@375 | 483 autorewrite with core in *; simpl in *; intuition; |

adam@375 | 484 subst; autorewrite with core in *; simpl in *; |

adamc@240 | 485 intuition; subst. |

adamc@240 | 486 |

adamc@240 | 487 ]] |

adamc@240 | 488 |

adamc@240 | 489 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 | 490 |

adamc@240 | 491 Undo. |

adamc@240 | 492 |

adamc@240 | 493 (** We arbitrarily split the script into chunks. The first few seem not to do any harm. *) |

adamc@240 | 494 |

adam@375 | 495 simpl in *; intuition; subst; autorewrite with core in *. |

adam@375 | 496 simpl in *; intuition; subst; autorewrite with core in *. |

adamc@240 | 497 simpl in *; intuition; subst; destruct (reassoc e2). |

adamc@240 | 498 simpl in *; intuition. |

adamc@240 | 499 simpl in *; intuition. |

adamc@240 | 500 |

adamc@240 | 501 (** The next step is revealed as the culprit, bringing us to the final unproved subgoal. *) |

adamc@240 | 502 |

adam@375 | 503 simpl in *; intuition; subst; autorewrite with core in *. |

adamc@240 | 504 |

adamc@240 | 505 (** We can split the steps further to assign blame. *) |

adamc@240 | 506 |

adamc@240 | 507 Undo. |

adamc@240 | 508 |

adamc@240 | 509 simpl in *. |

adamc@240 | 510 intuition. |

adamc@240 | 511 subst. |

adam@375 | 512 autorewrite with core in *. |

adamc@240 | 513 |

adamc@240 | 514 (** 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 | 515 |

adamc@240 | 516 Undo. |

adamc@240 | 517 |

adam@375 | 518 info autorewrite with core in *. |

adam@367 | 519 (** %\vspace{-.15in}%[[ |

adamc@240 | 520 == refine (eq_ind_r (fun n : nat => n = eval e1 * eval e2) _ |

adamc@240 | 521 (confounder (reassoc e1) e3 e4)). |

adamc@240 | 522 ]] |

adamc@240 | 523 |

adamc@240 | 524 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 | 525 |

adamc@240 | 526 Abort. |

adam@368 | 527 (* end thide *) |

adamc@240 | 528 |

adamc@241 | 529 (** 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 | 530 |

adamc@241 | 531 Here is one example of a performance surprise. *) |

adamc@241 | 532 |

adamc@239 | 533 Section slow. |

adamc@239 | 534 Hint Resolve trans_eq. |

adamc@239 | 535 |

adamc@241 | 536 (** 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 | 537 |

adamc@239 | 538 Variable A : Set. |

adamc@239 | 539 Variables P Q R S : A -> A -> Prop. |

adamc@239 | 540 Variable f : A -> A. |

adamc@239 | 541 |

adamc@239 | 542 Hypothesis H1 : forall x y, P x y -> Q x y -> R x y -> f x = f y. |

adamc@239 | 543 Hypothesis H2 : forall x y, S x y -> R x y. |

adamc@239 | 544 |

adam@367 | 545 (** We prove a simple lemma very quickly, using the %\index{Vernacular commands!Time}%[Time] command to measure exactly how quickly. *) |

adamc@241 | 546 |

adamc@239 | 547 Lemma slow : forall x y, P x y -> Q x y -> S x y -> f x = f y. |

adamc@241 | 548 Time eauto 6. |

adam@433 | 549 (** << |

adamc@241 | 550 Finished transaction in 0. secs (0.068004u,0.s) |

adam@433 | 551 >> |

adam@302 | 552 *) |

adamc@241 | 553 |

adamc@239 | 554 Qed. |

adamc@239 | 555 |

adamc@241 | 556 (** Now we add a different hypothesis, which is innocent enough; in fact, it is even provable as a theorem. *) |

adamc@241 | 557 |

adamc@239 | 558 Hypothesis H3 : forall x y, x = y -> f x = f y. |

adamc@239 | 559 |

adamc@239 | 560 Lemma slow' : forall x y, P x y -> Q x y -> S x y -> f x = f y. |

adamc@241 | 561 Time eauto 6. |

adam@433 | 562 (** << |

adamc@241 | 563 Finished transaction in 2. secs (1.264079u,0.s) |

adam@433 | 564 >> |

adam@445 | 565 %\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 | 566 |

adam@368 | 567 (* begin thide *) |

adamc@241 | 568 Restart. |

adamc@241 | 569 info eauto 6. |

adam@367 | 570 (** %\vspace{-.15in}%[[ |

adamc@241 | 571 == intro x; intro y; intro H; intro H0; intro H4; |

adamc@241 | 572 simple eapply trans_eq. |

adam@426 | 573 simple apply eq_refl. |

adamc@241 | 574 |

adamc@241 | 575 simple eapply trans_eq. |

adam@426 | 576 simple apply eq_refl. |

adamc@241 | 577 |

adamc@241 | 578 simple eapply trans_eq. |

adam@426 | 579 simple apply eq_refl. |

adamc@241 | 580 |

adamc@241 | 581 simple apply H1. |

adamc@241 | 582 eexact H. |

adamc@241 | 583 |

adamc@241 | 584 eexact H0. |

adamc@241 | 585 |

adamc@241 | 586 simple apply H2; eexact H4. |

adamc@241 | 587 ]] |

adamc@241 | 588 |

adam@367 | 589 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 | 590 |

adamc@241 | 591 Restart. |

adamc@241 | 592 debug eauto 6. |

adamc@241 | 593 |

adam@433 | 594 (* begin hide *) |

adam@437 | 595 (* begin thide *) |

adam@433 | 596 Definition deeeebug := (@eq_refl, @sym_eq). |

adam@437 | 597 (* end thide *) |

adam@433 | 598 (* end hide *) |

adam@433 | 599 |

adamc@241 | 600 (** The output is a large proof tree. The beginning of the tree is enough to reveal what is happening: |

adamc@241 | 601 [[ |

adamc@241 | 602 1 depth=6 |

adamc@241 | 603 1.1 depth=6 intro |

adamc@241 | 604 1.1.1 depth=6 intro |

adamc@241 | 605 1.1.1.1 depth=6 intro |

adamc@241 | 606 1.1.1.1.1 depth=6 intro |

adamc@241 | 607 1.1.1.1.1.1 depth=6 intro |

adamc@241 | 608 1.1.1.1.1.1.1 depth=5 apply H3 |

adamc@241 | 609 1.1.1.1.1.1.1.1 depth=4 eapply trans_eq |

adam@426 | 610 1.1.1.1.1.1.1.1.1 depth=4 apply eq_refl |

adamc@241 | 611 1.1.1.1.1.1.1.1.1.1 depth=3 eapply trans_eq |

adam@426 | 612 1.1.1.1.1.1.1.1.1.1.1 depth=3 apply eq_refl |

adamc@241 | 613 1.1.1.1.1.1.1.1.1.1.1.1 depth=2 eapply trans_eq |

adam@426 | 614 1.1.1.1.1.1.1.1.1.1.1.1.1 depth=2 apply eq_refl |

adamc@241 | 615 1.1.1.1.1.1.1.1.1.1.1.1.1.1 depth=1 eapply trans_eq |

adam@426 | 616 1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 depth=1 apply eq_refl |

adamc@241 | 617 1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 depth=0 eapply trans_eq |

adamc@241 | 618 1.1.1.1.1.1.1.1.1.1.1.1.1.1.2 depth=1 apply sym_eq ; trivial |

adamc@241 | 619 1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1 depth=0 eapply trans_eq |

adamc@241 | 620 1.1.1.1.1.1.1.1.1.1.1.1.1.1.3 depth=0 eapply trans_eq |

adamc@241 | 621 1.1.1.1.1.1.1.1.1.1.1.1.2 depth=2 apply sym_eq ; trivial |

adamc@241 | 622 1.1.1.1.1.1.1.1.1.1.1.1.2.1 depth=1 eapply trans_eq |

adam@426 | 623 1.1.1.1.1.1.1.1.1.1.1.1.2.1.1 depth=1 apply eq_refl |

adamc@241 | 624 1.1.1.1.1.1.1.1.1.1.1.1.2.1.1.1 depth=0 eapply trans_eq |

adamc@241 | 625 1.1.1.1.1.1.1.1.1.1.1.1.2.1.2 depth=1 apply sym_eq ; trivial |

adamc@241 | 626 1.1.1.1.1.1.1.1.1.1.1.1.2.1.2.1 depth=0 eapply trans_eq |

adamc@241 | 627 1.1.1.1.1.1.1.1.1.1.1.1.2.1.3 depth=0 eapply trans_eq |

adamc@241 | 628 ]] |

adamc@241 | 629 |

adam@367 | 630 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 | 631 |

adamc@239 | 632 Qed. |

adam@368 | 633 (* end thide *) |

adamc@239 | 634 End slow. |

adamc@239 | 635 |

adam@387 | 636 (** 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 | 637 |

adam@413 | 638 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 | 639 |

adam@433 | 640 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 | 641 |

adamc@238 | 642 |

adamc@235 | 643 (** * Modules *) |

adamc@235 | 644 |

adam@462 | 645 (** 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 | 646 |

adam@413 | 647 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 | 648 |

adam@462 | 649 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 | 650 |

adamc@235 | 651 Module Type GROUP. |

adamc@235 | 652 Parameter G : Set. |

adamc@235 | 653 Parameter f : G -> G -> G. |

adam@462 | 654 Parameter id : G. |

adamc@235 | 655 Parameter i : G -> G. |

adamc@235 | 656 |

adamc@235 | 657 Axiom assoc : forall a b c, f (f a b) c = f a (f b c). |

adam@462 | 658 Axiom ident : forall a, f id a = a. |

adam@462 | 659 Axiom inverse : forall a, f (i a) a = id. |

adamc@235 | 660 End GROUP. |

adamc@235 | 661 |

adam@367 | 662 (** Many useful theorems hold of arbitrary groups. We capture some such theorem statements in another module signature.%\index{Vernacular commands!Declare Module}% *) |

adamc@242 | 663 |

adamc@235 | 664 Module Type GROUP_THEOREMS. |

adamc@235 | 665 Declare Module M : GROUP. |

adamc@235 | 666 |

adam@462 | 667 Axiom ident' : forall a, M.f a M.id = a. |

adamc@235 | 668 |

adam@462 | 669 Axiom inverse' : forall a, M.f a (M.i a) = M.id. |

adamc@235 | 670 |

adam@462 | 671 Axiom unique_ident : forall id', (forall a, M.f id' a = a) -> id' = M.id. |

adamc@235 | 672 End GROUP_THEOREMS. |

adamc@235 | 673 |

adam@387 | 674 (** We implement generic proofs of these theorems with a functor, whose input is an arbitrary group [M]. %\index{Vernacular commands!Module}% *) |

adamc@242 | 675 |

adam@387 | 676 Module GroupProofs (M : GROUP) : GROUP_THEOREMS with Module M := M. |

adam@509 | 677 |

adam@413 | 678 (** 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 | 679 |

adamc@235 | 680 Module M := M. |

adam@387 | 681 (** 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 | 682 |

adamc@235 | 683 Import M. |

adam@387 | 684 (** 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 | 685 |

adam@387 | 686 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 | 687 |

adam@462 | 688 Theorem inverse' : forall a, f a (i a) = id. |

adamc@235 | 689 intro. |

adamc@235 | 690 rewrite <- (ident (f a (i a))). |

adamc@235 | 691 rewrite <- (inverse (f a (i a))) at 1. |

adamc@235 | 692 rewrite assoc. |

adamc@235 | 693 rewrite assoc. |

adamc@235 | 694 rewrite <- (assoc (i a) a (i a)). |

adamc@235 | 695 rewrite inverse. |

adamc@235 | 696 rewrite ident. |

adamc@235 | 697 apply inverse. |

adamc@235 | 698 Qed. |

adamc@235 | 699 |

adam@462 | 700 Theorem ident' : forall a, f a id = a. |

adamc@235 | 701 intro. |

adamc@235 | 702 rewrite <- (inverse a). |

adamc@235 | 703 rewrite <- assoc. |

adamc@235 | 704 rewrite inverse'. |

adamc@235 | 705 apply ident. |

adamc@235 | 706 Qed. |

adamc@235 | 707 |

adam@462 | 708 Theorem unique_ident : forall id', (forall a, M.f id' a = a) -> id' = M.id. |

adamc@235 | 709 intros. |

adam@462 | 710 rewrite <- (H id). |

adamc@235 | 711 symmetry. |

adamc@235 | 712 apply ident'. |

adamc@235 | 713 Qed. |

adam@387 | 714 End GroupProofs. |

adamc@239 | 715 |

adamc@242 | 716 (** We can show that the integers with [+] form a group. *) |

adamc@242 | 717 |

adamc@239 | 718 Require Import ZArith. |

adamc@239 | 719 Open Scope Z_scope. |

adamc@239 | 720 |

adamc@239 | 721 Module Int. |

adamc@239 | 722 Definition G := Z. |

adamc@239 | 723 Definition f x y := x + y. |

adam@462 | 724 Definition id := 0. |

adamc@239 | 725 Definition i x := -x. |

adamc@239 | 726 |

adamc@239 | 727 Theorem assoc : forall a b c, f (f a b) c = f a (f b c). |

adamc@239 | 728 unfold f; crush. |

adamc@239 | 729 Qed. |

adam@462 | 730 Theorem ident : forall a, f id a = a. |

adam@462 | 731 unfold f, id; crush. |

adamc@239 | 732 Qed. |

adam@462 | 733 Theorem inverse : forall a, f (i a) a = id. |

adam@462 | 734 unfold f, i, id; crush. |

adamc@239 | 735 Qed. |

adamc@239 | 736 End Int. |

adamc@239 | 737 |

adamc@242 | 738 (** Next, we can produce integer-specific versions of the generic group theorems. *) |

adamc@242 | 739 |

adam@387 | 740 Module IntProofs := GroupProofs(Int). |

adamc@239 | 741 |

adam@387 | 742 Check IntProofs.unique_ident. |

adamc@242 | 743 (** %\vspace{-.15in}% [[ |

adam@387 | 744 IntProofs.unique_ident |

adamc@242 | 745 : forall e' : Int.G, (forall a : Int.G, Int.f e' a = a) -> e' = Int.e |

adam@302 | 746 ]] |

adam@367 | 747 |

adam@367 | 748 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 | 749 |

adam@462 | 750 Theorem unique_ident : forall id', (forall a, id' + a = a) -> id' = 0. |

adam@368 | 751 (* begin thide *) |

adam@387 | 752 exact IntProofs.unique_ident. |

adamc@239 | 753 Qed. |

adam@368 | 754 (* end thide *) |

adamc@242 | 755 |

adam@475 | 756 (** 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 cases. 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 | 757 |

adamc@243 | 758 |

adamc@243 | 759 (** * Build Processes *) |

adamc@243 | 760 |

adam@433 | 761 (* begin hide *) |

adam@437 | 762 (* begin thide *) |

adam@433 | 763 Module Lib. |

adam@433 | 764 Module A. |

adam@433 | 765 End A. |

adam@433 | 766 Module B. |

adam@433 | 767 End B. |

adam@433 | 768 Module C. |

adam@433 | 769 End C. |

adam@433 | 770 End Lib. |

adam@433 | 771 Module Client. |

adam@433 | 772 Module D. |

adam@433 | 773 End D. |

adam@433 | 774 Module E. |

adam@433 | 775 End E. |

adam@433 | 776 End Client. |

adam@437 | 777 (* end thide *) |

adam@433 | 778 (* end hide *) |

adam@433 | 779 |

adamc@243 | 780 (** 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 | 781 |

adam@435 | 782 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 | 783 |

adamc@243 | 784 << |

adamc@243 | 785 MODULES := A B C |

adamc@243 | 786 VS := $(MODULES:%=%.v) |

adamc@243 | 787 |

adamc@243 | 788 .PHONY: coq clean |

adamc@243 | 789 |

adamc@243 | 790 coq: Makefile.coq |

adam@369 | 791 $(MAKE) -f Makefile.coq |

adamc@243 | 792 |

adamc@243 | 793 Makefile.coq: Makefile $(VS) |

adamc@243 | 794 coq_makefile -R . Lib $(VS) -o Makefile.coq |

adamc@243 | 795 |

adamc@243 | 796 clean:: Makefile.coq |

adam@369 | 797 $(MAKE) -f Makefile.coq clean |

adamc@243 | 798 rm -f Makefile.coq |

adamc@243 | 799 >> |

adamc@243 | 800 |

adam@435 | 801 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 | 802 |

adam@433 | 803 Now code in <<B.v>> may refer to definitions in <<A.v>> after running |

adamc@243 | 804 [[ |

adamc@243 | 805 Require Import Lib.A. |

adam@367 | 806 ]] |

adam@433 | 807 %\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 | 808 |

adam@433 | 809 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 | 810 |

adam@433 | 811 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 | 812 |

adamc@243 | 813 << |

adamc@243 | 814 MODULES := D E |

adamc@243 | 815 VS := $(MODULES:%=%.v) |

adamc@243 | 816 |

adamc@243 | 817 .PHONY: coq clean |

adamc@243 | 818 |

adamc@243 | 819 coq: Makefile.coq |

adam@369 | 820 $(MAKE) -f Makefile.coq |

adamc@243 | 821 |

adamc@243 | 822 Makefile.coq: Makefile $(VS) |

adamc@243 | 823 coq_makefile -R LIB Lib -R . Client $(VS) -o Makefile.coq |

adamc@243 | 824 |

adamc@243 | 825 clean:: Makefile.coq |

adam@369 | 826 $(MAKE) -f Makefile.coq clean |

adamc@243 | 827 rm -f Makefile.coq |

adamc@243 | 828 >> |

adamc@243 | 829 |

adam@435 | 830 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 | 831 [[ |

adamc@243 | 832 Require Import Lib.A. |

adamc@243 | 833 ]] |

adam@433 | 834 %\vspace{-.15in}\noindent{}%and <<E.v>> can refer to definitions from <<D.v>> by running |

adamc@243 | 835 [[ |

adamc@243 | 836 Require Import Client.D. |

adamc@243 | 837 ]] |

adam@433 | 838 %\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 | 839 [[ |

adamc@243 | 840 Require Export Lib.A Lib.B Lib.C. |

adamc@243 | 841 ]] |

adam@367 | 842 %\vspace{-.15in}%Now client code can import all definitions from all of [Lib]'s modules simply by running |

adamc@243 | 843 [[ |

adamc@243 | 844 Require Import Lib. |

adamc@243 | 845 ]] |

adam@367 | 846 %\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 | 847 |

adamc@243 | 848 %\medskip% |

adamc@243 | 849 |

adam@433 | 850 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 | 851 |

adamc@243 | 852 << |

adamc@243 | 853 (custom-set-variables |

adamc@243 | 854 ... |

adam@535 | 855 '(coq-prog-args '("-R" "/path/to/cpdt/src" "Cpdt")) |

adamc@243 | 856 ... |

adamc@243 | 857 ) |

adamc@243 | 858 >> |

adamc@243 | 859 |

adamc@243 | 860 To do interactive editing of our current example, we just need to change the flags to point to the right places. |

adamc@243 | 861 |

adamc@243 | 862 << |

adamc@243 | 863 (custom-set-variables |

adamc@243 | 864 ... |

adam@535 | 865 ; '(coq-prog-args '("-R" "/path/to/cpdt/src" "Cpdt")) |

adamc@243 | 866 '(coq-prog-args '("-R" "LIB" "Lib" "-R" "CLIENT" "Client")) |

adamc@243 | 867 ... |

adamc@243 | 868 ) |

adamc@243 | 869 >> |

adamc@243 | 870 |

adam@433 | 871 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 | 872 |

adam@433 | 873 Alternatively, we can revisit the directory-local settings approach and write the following into a file <<.dir-locals.el>> in <<CLIENT>>: |

adam@397 | 874 |

adam@397 | 875 << |

adam@397 | 876 ((coq-mode . ((coq-prog-args . |

adam@397 | 877 ("-emacs-U" "-R" "LIB" "Lib" "-R" "CLIENT" "Client"))))) |

adam@397 | 878 >> |

adam@509 | 879 |

adam@509 | 880 A downside of this approach is that users of your code may not want to trust the arbitrary Emacs Lisp programs that you are allowed to place in such files, so that they prefer to add mappings manually. |

adam@397 | 881 *) |