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Prose for Modules section
author Adam Chlipala <adamc@hcoop.net>
date Wed, 09 Dec 2009 13:07:31 -0500
parents 82eae7bc91ea
children b653e6b19b6d
rev   line source
adamc@220 1 (* Copyright (c) 2008-2009, Adam Chlipala
adamc@132 2 *
adamc@132 3 * This work is licensed under a
adamc@132 4 * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
adamc@132 5 * Unported License.
adamc@132 6 * The license text is available at:
adamc@132 7 * http://creativecommons.org/licenses/by-nc-nd/3.0/
adamc@132 8 *)
adamc@132 9
adamc@132 10 (* begin hide *)
adamc@132 11 Require Import List.
adamc@132 12
adamc@132 13 Require Import Tactics.
adamc@132 14
adamc@132 15 Set Implicit Arguments.
adamc@132 16 (* end hide *)
adamc@132 17
adamc@132 18
adamc@132 19 (** %\part{Proof Engineering}
adamc@132 20
adamc@132 21 \chapter{Proof Search in Ltac}% *)
adamc@132 22
adamc@132 23 (** We have seen many examples of proof automation so far. This chapter aims to give a principled presentation of the features of Ltac, focusing in particular on the Ltac [match] construct, which supports a novel approach to backtracking search. First, though, we will run through some useful automation tactics that are built into Coq. They are described in detail in the manual, so we only outline what is possible. *)
adamc@132 24
adamc@132 25 (** * Some Built-In Automation Tactics *)
adamc@132 26
adamc@132 27 (** A number of tactics are called repeatedly by [crush]. [intuition] simplifies propositional structure of goals. [congruence] applies the rules of equality and congruence closure, plus properties of constructors of inductive types. The [omega] tactic provides a complete decision procedure for a theory that is called quantifier-free linear arithmetic or Presburger arithmetic, depending on whom you ask. That is, [omega] proves any goal that follows from looking only at parts of that goal that can be interpreted as propositional formulas whose atomic formulas are basic comparison operations on natural numbers or integers.
adamc@132 28
adamc@132 29 The [ring] tactic solves goals by appealing to the axioms of rings or semi-rings (as in algebra), depending on the type involved. Coq developments may declare new types to be parts of rings and semi-rings by proving the associated axioms. There is a simlar tactic [field] for simplifying values in fields by conversion to fractions over rings. Both [ring] and [field] can only solve goals that are equalities. The [fourier] tactic uses Fourier's method to prove inequalities over real numbers, which are axiomatized in the Coq standard library.
adamc@132 30
adamc@133 31 The %\textit{%#<i>#setoid#</i>#%}% facility makes it possible to register new equivalence relations to be understood by tactics like [rewrite]. For instance, [Prop] is registered as a setoid with the equivalence relation "if and only if." The ability to register new setoids can be very useful in proofs of a kind common in math, where all reasoning is done after "modding out by a relation." *)
adamc@132 32
adamc@132 33
adamc@133 34 (** * Hint Databases *)
adamc@132 35
adamc@133 36 (** Another class of built-in tactics includes [auto], [eauto], and [autorewrite]. These are based on %\textit{%#<i>#hint databases#</i>#%}%, which we have seen extended in many examples so far. These tactics are important, because, in Ltac programming, we cannot create "global variables" whose values can be extended seamlessly by different modules in different source files. We have seen the advantages of hints so far, where [crush] can be defined once and for all, while still automatically applying the hints we add throughout developments.
adamc@133 37
adamc@220 38 The basic hints for [auto] and [eauto] are [Hint Immediate lemma], asking to try solving a goal immediately by applying a lemma and discharging any hypotheses with a single proof step each; [Resolve lemma], which does the same but may add new premises that are themselves to be subjects of nested proof search; [Constructor type], which acts like [Resolve] applied to every constructor of an inductive type; and [Unfold ident], which tries unfolding [ident] when it appears at the head of a proof goal. Each of these [Hint] commands may be used with a suffix, as in [Hint Resolve lemma : my_db]. This adds the hint only to the specified database, so that it would only be used by, for instance, [auto with my_db]. An additional argument to [auto] specifies the maximum depth of proof trees to search in depth-first order, as in [auto 8] or [auto 8 with my_db]. The default depth is 5.
adamc@133 39
adamc@133 40 All of these [Hint] commands can be issued alternatively with a more primitive hint kind, [Extern]. A few examples should do best to explain how [Hint Extern] works. *)
adamc@133 41
adamc@133 42 Theorem bool_neq : true <> false.
adamc@141 43 (* begin thide *)
adamc@133 44 auto.
adamc@220 45
adamc@133 46 (** [crush] would have discharged this goal, but the default hint database for [auto] contains no hint that applies. *)
adamc@220 47
adamc@133 48 Abort.
adamc@133 49
adamc@133 50 (** It is hard to come up with a [bool]-specific hint that is not just a restatement of the theorem we mean to prove. Luckily, a simpler form suffices. *)
adamc@133 51
adamc@133 52 Hint Extern 1 (_ <> _) => congruence.
adamc@133 53
adamc@133 54 Theorem bool_neq : true <> false.
adamc@133 55 auto.
adamc@133 56 Qed.
adamc@141 57 (* end thide *)
adamc@133 58
adamc@133 59 (** Our hint says: "whenever the conclusion matches the pattern [_ <> _], try applying [congruence]." The [1] is a cost for this rule. During proof search, whenever multiple rules apply, rules are tried in increasing cost order, so it pays to assign high costs to relatively expensive [Extern] hints.
adamc@133 60
adamc@133 61 [Extern] hints may be implemented with the full Ltac language. This example shows a case where a hint uses a [match]. *)
adamc@133 62
adamc@133 63 Section forall_and.
adamc@133 64 Variable A : Set.
adamc@133 65 Variables P Q : A -> Prop.
adamc@133 66
adamc@133 67 Hypothesis both : forall x, P x /\ Q x.
adamc@133 68
adamc@133 69 Theorem forall_and : forall z, P z.
adamc@141 70 (* begin thide *)
adamc@133 71 crush.
adamc@220 72
adamc@133 73 (** [crush] makes no progress beyond what [intros] would have accomplished. [auto] will not apply the hypothesis [both] to prove the goal, because the conclusion of [both] does not unify with the conclusion of the goal. However, we can teach [auto] to handle this kind of goal. *)
adamc@133 74
adamc@133 75 Hint Extern 1 (P ?X) =>
adamc@133 76 match goal with
adamc@133 77 | [ H : forall x, P x /\ _ |- _ ] => apply (proj1 (H X))
adamc@133 78 end.
adamc@133 79
adamc@133 80 auto.
adamc@133 81 Qed.
adamc@141 82 (* end thide *)
adamc@133 83
adamc@133 84 (** We see that an [Extern] pattern may bind unification variables that we use in the associated tactic. [proj1] is a function from the standard library for extracting a proof of [R] from a proof of [R /\ S]. *)
adamc@220 85
adamc@133 86 End forall_and.
adamc@133 87
adamc@133 88 (** After our success on this example, we might get more ambitious and seek to generalize the hint to all possible predicates [P].
adamc@133 89
adamc@133 90 [[
adamc@133 91 Hint Extern 1 (?P ?X) =>
adamc@133 92 match goal with
adamc@220 93 | [ H : forall x, P x /\ _ |- _ ] => apply (proj1 (H X))
adamc@133 94 end.
adamc@133 95
adamc@133 96 User error: Bound head variable
adamc@220 97
adamc@134 98 ]]
adamc@133 99
adamc@134 100 Coq's [auto] hint databases work as tables mapping %\textit{%#<i>#head symbols#</i>#%}% to lists of tactics to try. Because of this, the constant head of an [Extern] pattern must be determinable statically. In our first [Extern] hint, the head symbol was [not], since [x <> y] desugars to [not (eq x y)]; and, in the second example, the head symbol was [P].
adamc@133 101
adamc@134 102 This restriction on [Extern] hints is the main limitation of the [auto] mechanism, preventing us from using it for general context simplifications that are not keyed off of the form of the conclusion. This is perhaps just as well, since we can often code more efficient tactics with specialized Ltac programs, and we will see how in later sections of the chapter.
adamc@134 103
adamc@134 104 We have used [Hint Rewrite] in many examples so far. [crush] uses these hints by calling [autorewrite]. Our rewrite hints have taken the form [Hint Rewrite lemma : cpdt], adding them to the [cpdt] rewrite database. This is because, in contrast to [auto], [autorewrite] has no default database. Thus, we set the convention that [crush] uses the [cpdt] database.
adamc@134 105
adamc@134 106 This example shows a direct use of [autorewrite]. *)
adamc@134 107
adamc@134 108 Section autorewrite.
adamc@134 109 Variable A : Set.
adamc@134 110 Variable f : A -> A.
adamc@134 111
adamc@134 112 Hypothesis f_f : forall x, f (f x) = f x.
adamc@134 113
adamc@134 114 Hint Rewrite f_f : my_db.
adamc@134 115
adamc@134 116 Lemma f_f_f : forall x, f (f (f x)) = f x.
adamc@134 117 intros; autorewrite with my_db; reflexivity.
adamc@134 118 Qed.
adamc@134 119
adamc@134 120 (** There are a few ways in which [autorewrite] can lead to trouble when insufficient care is taken in choosing hints. First, the set of hints may define a nonterminating rewrite system, in which case invocations to [autorewrite] may not terminate. Second, we may add hints that "lead [autorewrite] down the wrong path." For instance: *)
adamc@134 121
adamc@134 122 Section garden_path.
adamc@134 123 Variable g : A -> A.
adamc@134 124 Hypothesis f_g : forall x, f x = g x.
adamc@134 125 Hint Rewrite f_g : my_db.
adamc@134 126
adamc@134 127 Lemma f_f_f' : forall x, f (f (f x)) = f x.
adamc@134 128 intros; autorewrite with my_db.
adamc@134 129 (** [[
adamc@134 130 ============================
adamc@134 131 g (g (g x)) = g x
adamc@134 132 ]] *)
adamc@220 133
adamc@134 134 Abort.
adamc@134 135
adamc@134 136 (** Our new hint was used to rewrite the goal into a form where the old hint could no longer be applied. This "non-monotonicity" of rewrite hints contrasts with the situation for [auto], where new hints may slow down proof search but can never "break" old proofs. *)
adamc@134 137
adamc@134 138 Reset garden_path.
adamc@134 139
adamc@134 140 (** [autorewrite] works with quantified equalities that include additional premises, but we must be careful to avoid similar incorrect rewritings. *)
adamc@134 141
adamc@134 142 Section garden_path.
adamc@134 143 Variable P : A -> Prop.
adamc@134 144 Variable g : A -> A.
adamc@134 145 Hypothesis f_g : forall x, P x -> f x = g x.
adamc@134 146 Hint Rewrite f_g : my_db.
adamc@134 147
adamc@134 148 Lemma f_f_f' : forall x, f (f (f x)) = f x.
adamc@134 149 intros; autorewrite with my_db.
adamc@134 150 (** [[
adamc@134 151
adamc@134 152 ============================
adamc@134 153 g (g (g x)) = g x
adamc@134 154
adamc@134 155 subgoal 2 is:
adamc@134 156 P x
adamc@134 157 subgoal 3 is:
adamc@134 158 P (f x)
adamc@134 159 subgoal 4 is:
adamc@134 160 P (f x)
adamc@134 161 ]] *)
adamc@220 162
adamc@134 163 Abort.
adamc@134 164
adamc@134 165 (** The inappropriate rule fired the same three times as before, even though we know we will not be able to prove the premises. *)
adamc@134 166
adamc@134 167 Reset garden_path.
adamc@134 168
adamc@134 169 (** Our final, successful, attempt uses an extra argument to [Hint Rewrite] that specifies a tactic to apply to generated premises. *)
adamc@134 170
adamc@134 171 Section garden_path.
adamc@134 172 Variable P : A -> Prop.
adamc@134 173 Variable g : A -> A.
adamc@134 174 Hypothesis f_g : forall x, P x -> f x = g x.
adamc@141 175 (* begin thide *)
adamc@134 176 Hint Rewrite f_g using assumption : my_db.
adamc@141 177 (* end thide *)
adamc@134 178
adamc@134 179 Lemma f_f_f' : forall x, f (f (f x)) = f x.
adamc@141 180 (* begin thide *)
adamc@134 181 intros; autorewrite with my_db; reflexivity.
adamc@134 182 Qed.
adamc@141 183 (* end thide *)
adamc@134 184
adamc@134 185 (** [autorewrite] will still use [f_g] when the generated premise is among our assumptions. *)
adamc@134 186
adamc@134 187 Lemma f_f_f_g : forall x, P x -> f (f x) = g x.
adamc@141 188 (* begin thide *)
adamc@134 189 intros; autorewrite with my_db; reflexivity.
adamc@141 190 (* end thide *)
adamc@134 191 Qed.
adamc@134 192 End garden_path.
adamc@134 193
adamc@220 194 (** remove printing * *)
adamc@220 195
adamc@134 196 (** It can also be useful to use the [autorewrite with db in *] form, which does rewriting in hypotheses, as well as in the conclusion. *)
adamc@134 197
adamc@220 198 (** printing * $*$ *)
adamc@220 199
adamc@134 200 Lemma in_star : forall x y, f (f (f (f x))) = f (f y)
adamc@134 201 -> f x = f (f (f y)).
adamc@141 202 (* begin thide *)
adamc@134 203 intros; autorewrite with my_db in *; assumption.
adamc@141 204 (* end thide *)
adamc@134 205 Qed.
adamc@134 206
adamc@134 207 End autorewrite.
adamc@135 208
adamc@135 209
adamc@135 210 (** * Ltac Programming Basics *)
adamc@135 211
adamc@135 212 (** We have already seen many examples of Ltac programs. In the rest of this chapter, we attempt to give a more principled introduction to the important features and design patterns.
adamc@135 213
adamc@135 214 One common use for [match] tactics is identification of subjects for case analysis, as we see in this tactic definition. *)
adamc@135 215
adamc@141 216 (* begin thide *)
adamc@135 217 Ltac find_if :=
adamc@135 218 match goal with
adamc@135 219 | [ |- if ?X then _ else _ ] => destruct X
adamc@135 220 end.
adamc@141 221 (* end thide *)
adamc@135 222
adamc@135 223 (** The tactic checks if the conclusion is an [if], [destruct]ing the test expression if so. Certain classes of theorem are trivial to prove automatically with such a tactic. *)
adamc@135 224
adamc@135 225 Theorem hmm : forall (a b c : bool),
adamc@135 226 if a
adamc@135 227 then if b
adamc@135 228 then True
adamc@135 229 else True
adamc@135 230 else if c
adamc@135 231 then True
adamc@135 232 else True.
adamc@141 233 (* begin thide *)
adamc@135 234 intros; repeat find_if; constructor.
adamc@135 235 Qed.
adamc@141 236 (* end thide *)
adamc@135 237
adamc@135 238 (** The [repeat] that we use here is called a %\textit{%#<i>#tactical#</i>#%}%, or tactic combinator. The behavior of [repeat t] is to loop through running [t], running [t] on all generated subgoals, running [t] on %\textit{%#<i>#their#</i>#%}% generated subgoals, and so on. When [t] fails at any point in this search tree, that particular subgoal is left to be handled by later tactics. Thus, it is important never to use [repeat] with a tactic that always succeeds.
adamc@135 239
adamc@135 240 Another very useful Ltac building block is %\textit{%#<i>#context patterns#</i>#%}%. *)
adamc@135 241
adamc@141 242 (* begin thide *)
adamc@135 243 Ltac find_if_inside :=
adamc@135 244 match goal with
adamc@135 245 | [ |- context[if ?X then _ else _] ] => destruct X
adamc@135 246 end.
adamc@141 247 (* end thide *)
adamc@135 248
adamc@135 249 (** The behavior of this tactic is to find any subterm of the conclusion that is an [if] and then [destruct] the test expression. This version subsumes [find_if]. *)
adamc@135 250
adamc@135 251 Theorem hmm' : forall (a b c : bool),
adamc@135 252 if a
adamc@135 253 then if b
adamc@135 254 then True
adamc@135 255 else True
adamc@135 256 else if c
adamc@135 257 then True
adamc@135 258 else True.
adamc@141 259 (* begin thide *)
adamc@135 260 intros; repeat find_if_inside; constructor.
adamc@135 261 Qed.
adamc@141 262 (* end thide *)
adamc@135 263
adamc@135 264 (** We can also use [find_if_inside] to prove goals that [find_if] does not simplify sufficiently. *)
adamc@135 265
adamc@141 266 Theorem hmm2 : forall (a b : bool),
adamc@135 267 (if a then 42 else 42) = (if b then 42 else 42).
adamc@141 268 (* begin thide *)
adamc@135 269 intros; repeat find_if_inside; reflexivity.
adamc@135 270 Qed.
adamc@141 271 (* end thide *)
adamc@135 272
adamc@135 273 (** Many decision procedures can be coded in Ltac via "[repeat match] loops." For instance, we can implement a subset of the functionality of [tauto]. *)
adamc@135 274
adamc@141 275 (* begin thide *)
adamc@135 276 Ltac my_tauto :=
adamc@135 277 repeat match goal with
adamc@135 278 | [ H : ?P |- ?P ] => exact H
adamc@135 279
adamc@135 280 | [ |- True ] => constructor
adamc@135 281 | [ |- _ /\ _ ] => constructor
adamc@135 282 | [ |- _ -> _ ] => intro
adamc@135 283
adamc@135 284 | [ H : False |- _ ] => destruct H
adamc@135 285 | [ H : _ /\ _ |- _ ] => destruct H
adamc@135 286 | [ H : _ \/ _ |- _ ] => destruct H
adamc@135 287
adamc@135 288 | [ H1 : ?P -> ?Q, H2 : ?P |- _ ] =>
adamc@135 289 let H := fresh "H" in
adamc@135 290 generalize (H1 H2); clear H1; intro H
adamc@135 291 end.
adamc@141 292 (* end thide *)
adamc@135 293
adamc@135 294 (** Since [match] patterns can share unification variables between hypothesis and conclusion patterns, it is easy to figure out when the conclusion matches a hypothesis. The [exact] tactic solves a goal completely when given a proof term of the proper type.
adamc@135 295
adamc@220 296 It is also trivial to implement the "introduction rules" for a few of the connectives. Implementing elimination rules is only a little more work, since we must give a name for a hypothesis to [destruct].
adamc@135 297
adamc@135 298 The last rule implements modus ponens. The most interesting part is the use of the Ltac-level [let] with a [fresh] expression. [fresh] takes in a name base and returns a fresh hypothesis variable based on that name. We use the new name variable [H] as the name we assign to the result of modus ponens. The use of [generalize] changes our conclusion to be an implication from [Q]. We clear the original hypothesis and move [Q] into the context with name [H]. *)
adamc@135 299
adamc@135 300 Section propositional.
adamc@135 301 Variables P Q R : Prop.
adamc@135 302
adamc@138 303 Theorem propositional : (P \/ Q \/ False) /\ (P -> Q) -> True /\ Q.
adamc@141 304 (* begin thide *)
adamc@135 305 my_tauto.
adamc@135 306 Qed.
adamc@141 307 (* end thide *)
adamc@135 308 End propositional.
adamc@135 309
adamc@135 310 (** It was relatively easy to implement modus ponens, because we do not lose information by clearing every implication that we use. If we want to implement a similarly-complete procedure for quantifier instantiation, we need a way to ensure that a particular proposition is not already included among our hypotheses. To do that effectively, we first need to learn a bit more about the semantics of [match].
adamc@135 311
adamc@135 312 It is tempting to assume that [match] works like it does in ML. In fact, there are a few critical differences in its behavior. One is that we may include arbitrary expressions in patterns, instead of being restricted to variables and constructors. Another is that the same variable may appear multiple times, inducing an implicit equality constraint.
adamc@135 313
adamc@135 314 There is a related pair of two other differences that are much more important than the others. [match] has a %\textit{%#<i>#backtracking semantics for failure#</i>#%}%. In ML, pattern matching works by finding the first pattern to match and then executing its body. If the body raises an exception, then the overall match raises the same exception. In Coq, failures in case bodies instead trigger continued search through the list of cases.
adamc@135 315
adamc@135 316 For instance, this (unnecessarily verbose) proof script works: *)
adamc@135 317
adamc@135 318 Theorem m1 : True.
adamc@135 319 match goal with
adamc@135 320 | [ |- _ ] => intro
adamc@135 321 | [ |- True ] => constructor
adamc@135 322 end.
adamc@141 323 (* begin thide *)
adamc@135 324 Qed.
adamc@141 325 (* end thide *)
adamc@135 326
adamc@135 327 (** The first case matches trivially, but its body tactic fails, since the conclusion does not begin with a quantifier or implication. In a similar ML match, that would mean that the whole pattern-match fails. In Coq, we backtrack and try the next pattern, which also matches. Its body tactic succeeds, so the overall tactic succeeds as well.
adamc@135 328
adamc@135 329 The example shows how failure can move to a different pattern within a [match]. Failure can also trigger an attempt to find %\textit{%#<i>#a different way of matching a single pattern#</i>#%}%. Consider another example: *)
adamc@135 330
adamc@135 331 Theorem m2 : forall P Q R : Prop, P -> Q -> R -> Q.
adamc@135 332 intros; match goal with
adamc@220 333 | [ H : _ |- _ ] => idtac H
adamc@135 334 end.
adamc@135 335
adamc@220 336 (** Coq prints "[H1]". By applying [idtac] with an argument, a convenient debugging tool for "leaking information out of [match]es," we see that this [match] first tries binding [H] to [H1], which cannot be used to prove [Q]. Nonetheless, the following variation on the tactic succeeds at proving the goal: *)
adamc@135 337
adamc@141 338 (* begin thide *)
adamc@135 339 match goal with
adamc@135 340 | [ H : _ |- _ ] => exact H
adamc@135 341 end.
adamc@135 342 Qed.
adamc@141 343 (* end thide *)
adamc@135 344
adamc@135 345 (** The tactic first unifies [H] with [H1], as before, but [exact H] fails in that case, so the tactic engine searches for more possible values of [H]. Eventually, it arrives at the correct value, so that [exact H] and the overall tactic succeed. *)
adamc@135 346
adamc@135 347 (** Now we are equipped to implement a tactic for checking that a proposition is not among our hypotheses: *)
adamc@135 348
adamc@141 349 (* begin thide *)
adamc@135 350 Ltac notHyp P :=
adamc@135 351 match goal with
adamc@135 352 | [ _ : P |- _ ] => fail 1
adamc@135 353 | _ =>
adamc@135 354 match P with
adamc@135 355 | ?P1 /\ ?P2 => first [ notHyp P1 | notHyp P2 | fail 2 ]
adamc@135 356 | _ => idtac
adamc@135 357 end
adamc@135 358 end.
adamc@141 359 (* end thide *)
adamc@135 360
adamc@135 361 (** We use the equality checking that is built into pattern-matching to see if there is a hypothesis that matches the proposition exactly. If so, we use the [fail] tactic. Without arguments, [fail] signals normal tactic failure, as you might expect. When [fail] is passed an argument [n], [n] is used to count outwards through the enclosing cases of backtracking search. In this case, [fail 1] says "fail not just in this pattern-matching branch, but for the whole [match]." The second case will never be tried when the [fail 1] is reached.
adamc@135 362
adamc@135 363 This second case, used when [P] matches no hypothesis, checks if [P] is a conjunction. Other simplifications may have split conjunctions into their component formulas, so we need to check that at least one of those components is also not represented. To achieve this, we apply the [first] tactical, which takes a list of tactics and continues down the list until one of them does not fail. The [fail 2] at the end says to [fail] both the [first] and the [match] wrapped around it.
adamc@135 364
adamc@135 365 The body of the [?P1 /\ ?P2] case guarantees that, if it is reached, we either succeed completely or fail completely. Thus, if we reach the wildcard case, [P] is not a conjunction. We use [idtac], a tactic that would be silly to apply on its own, since its effect is to succeed at doing nothing. Nonetheless, [idtac] is a useful placeholder for cases like what we see here.
adamc@135 366
adamc@135 367 With the non-presence check implemented, it is easy to build a tactic that takes as input a proof term and adds its conclusion as a new hypothesis, only if that conclusion is not already present, failing otherwise. *)
adamc@135 368
adamc@141 369 (* begin thide *)
adamc@135 370 Ltac extend pf :=
adamc@135 371 let t := type of pf in
adamc@135 372 notHyp t; generalize pf; intro.
adamc@141 373 (* end thide *)
adamc@135 374
adamc@135 375 (** We see the useful [type of] operator of Ltac. This operator could not be implemented in Gallina, but it is easy to support in Ltac. We end up with [t] bound to the type of [pf]. We check that [t] is not already present. If so, we use a [generalize]/[intro] combo to add a new hypothesis proved by [pf].
adamc@135 376
adamc@135 377 With these tactics defined, we can write a tactic [completer] for adding to the context all consequences of a set of simple first-order formulas. *)
adamc@135 378
adamc@141 379 (* begin thide *)
adamc@135 380 Ltac completer :=
adamc@135 381 repeat match goal with
adamc@135 382 | [ |- _ /\ _ ] => constructor
adamc@135 383 | [ H : _ /\ _ |- _ ] => destruct H
adamc@135 384 | [ H : ?P -> ?Q, H' : ?P |- _ ] =>
adamc@135 385 generalize (H H'); clear H; intro H
adamc@135 386 | [ |- forall x, _ ] => intro
adamc@135 387
adamc@135 388 | [ H : forall x, ?P x -> _, H' : ?P ?X |- _ ] =>
adamc@135 389 extend (H X H')
adamc@135 390 end.
adamc@141 391 (* end thide *)
adamc@135 392
adamc@135 393 (** We use the same kind of conjunction and implication handling as previously. Note that, since [->] is the special non-dependent case of [forall], the fourth rule handles [intro] for implications, too.
adamc@135 394
adamc@135 395 In the fifth rule, when we find a [forall] fact [H] with a premise matching one of our hypotheses, we add the appropriate instantiation of [H]'s conclusion, if we have not already added it.
adamc@135 396
adamc@135 397 We can check that [completer] is working properly: *)
adamc@135 398
adamc@135 399 Section firstorder.
adamc@135 400 Variable A : Set.
adamc@135 401 Variables P Q R S : A -> Prop.
adamc@135 402
adamc@135 403 Hypothesis H1 : forall x, P x -> Q x /\ R x.
adamc@135 404 Hypothesis H2 : forall x, R x -> S x.
adamc@135 405
adamc@135 406 Theorem fo : forall x, P x -> S x.
adamc@141 407 (* begin thide *)
adamc@135 408 completer.
adamc@135 409 (** [[
adamc@135 410 x : A
adamc@135 411 H : P x
adamc@135 412 H0 : Q x
adamc@135 413 H3 : R x
adamc@135 414 H4 : S x
adamc@135 415 ============================
adamc@135 416 S x
adamc@135 417 ]] *)
adamc@135 418
adamc@135 419 assumption.
adamc@135 420 Qed.
adamc@141 421 (* end thide *)
adamc@135 422 End firstorder.
adamc@135 423
adamc@135 424 (** We narrowly avoided a subtle pitfall in our definition of [completer]. Let us try another definition that even seems preferable to the original, to the untrained eye. *)
adamc@135 425
adamc@141 426 (* begin thide *)
adamc@135 427 Ltac completer' :=
adamc@135 428 repeat match goal with
adamc@135 429 | [ |- _ /\ _ ] => constructor
adamc@135 430 | [ H : _ /\ _ |- _ ] => destruct H
adamc@135 431 | [ H : ?P -> _, H' : ?P |- _ ] =>
adamc@135 432 generalize (H H'); clear H; intro H
adamc@135 433 | [ |- forall x, _ ] => intro
adamc@135 434
adamc@135 435 | [ H : forall x, ?P x -> _, H' : ?P ?X |- _ ] =>
adamc@135 436 extend (H X H')
adamc@135 437 end.
adamc@141 438 (* end thide *)
adamc@135 439
adamc@135 440 (** The only difference is in the modus ponens rule, where we have replaced an unused unification variable [?Q] with a wildcard. Let us try our example again with this version: *)
adamc@135 441
adamc@135 442 Section firstorder'.
adamc@135 443 Variable A : Set.
adamc@135 444 Variables P Q R S : A -> Prop.
adamc@135 445
adamc@135 446 Hypothesis H1 : forall x, P x -> Q x /\ R x.
adamc@135 447 Hypothesis H2 : forall x, R x -> S x.
adamc@135 448
adamc@135 449 Theorem fo' : forall x, P x -> S x.
adamc@141 450 (* begin thide *)
adamc@135 451 (** [[
adamc@135 452 completer'.
adamc@220 453
adamc@205 454 ]]
adamc@205 455
adamc@135 456 Coq loops forever at this point. What went wrong? *)
adamc@220 457
adamc@135 458 Abort.
adamc@141 459 (* end thide *)
adamc@135 460 End firstorder'.
adamc@136 461
adamc@136 462 (** A few examples should illustrate the issue. Here we see a [match]-based proof that works fine: *)
adamc@136 463
adamc@136 464 Theorem t1 : forall x : nat, x = x.
adamc@136 465 match goal with
adamc@136 466 | [ |- forall x, _ ] => trivial
adamc@136 467 end.
adamc@141 468 (* begin thide *)
adamc@136 469 Qed.
adamc@141 470 (* end thide *)
adamc@136 471
adamc@136 472 (** This one fails. *)
adamc@136 473
adamc@141 474 (* begin thide *)
adamc@136 475 Theorem t1' : forall x : nat, x = x.
adamc@136 476 (** [[
adamc@136 477 match goal with
adamc@136 478 | [ |- forall x, ?P ] => trivial
adamc@136 479 end.
adamc@136 480
adamc@136 481 User error: No matching clauses for match goal
adamc@136 482 ]] *)
adamc@220 483
adamc@136 484 Abort.
adamc@141 485 (* end thide *)
adamc@136 486
adamc@136 487 (** The problem is that unification variables may not contain locally-bound variables. In this case, [?P] would need to be bound to [x = x], which contains the local quantified variable [x]. By using a wildcard in the earlier version, we avoided this restriction.
adamc@136 488
adamc@136 489 The Coq 8.2 release includes a special pattern form for a unification variable with an explicit set of free variables. That unification variable is then bound to a function from the free variables to the "real" value. In Coq 8.1 and earlier, there is no such workaround.
adamc@136 490
adamc@136 491 No matter which version you use, it is important to be aware of this restriction. As we have alluded to, the restriction is the culprit behind the infinite-looping behavior of [completer']. We unintentionally match quantified facts with the modus ponens rule, circumventing the "already present" check and leading to different behavior. *)
adamc@137 492
adamc@137 493
adamc@137 494 (** * Functional Programming in Ltac *)
adamc@137 495
adamc@141 496 (* EX: Write a list length function in Ltac. *)
adamc@141 497
adamc@137 498 (** Ltac supports quite convenient functional programming, with a Lisp-with-syntax kind of flavor. However, there are a few syntactic conventions involved in getting programs to be accepted. The Ltac syntax is optimized for tactic-writing, so one has to deal with some inconveniences in writing more standard functional programs.
adamc@137 499
adamc@137 500 To illustrate, let us try to write a simple list length function. We start out writing it just like in Gallina, simply replacing [Fixpoint] (and its annotations) with [Ltac].
adamc@137 501
adamc@137 502 [[
adamc@137 503 Ltac length ls :=
adamc@137 504 match ls with
adamc@137 505 | nil => O
adamc@137 506 | _ :: ls' => S (length ls')
adamc@137 507 end.
adamc@137 508
adamc@137 509 Error: The reference ls' was not found in the current environment
adamc@220 510
adamc@137 511 ]]
adamc@137 512
adamc@137 513 At this point, we hopefully remember that pattern variable names must be prefixed by question marks in Ltac.
adamc@137 514
adamc@137 515 [[
adamc@137 516 Ltac length ls :=
adamc@137 517 match ls with
adamc@137 518 | nil => O
adamc@137 519 | _ :: ?ls' => S (length ls')
adamc@137 520 end.
adamc@137 521
adamc@137 522 Error: The reference S was not found in the current environment
adamc@220 523
adamc@137 524 ]]
adamc@137 525
adamc@137 526 The problem is that Ltac treats the expression [S (length ls')] as an invocation of a tactic [S] with argument [length ls']. We need to use a special annotation to "escape into" the Gallina parsing nonterminal. *)
adamc@137 527
adamc@141 528 (* begin thide *)
adamc@137 529 Ltac length ls :=
adamc@137 530 match ls with
adamc@137 531 | nil => O
adamc@137 532 | _ :: ?ls' => constr:(S (length ls'))
adamc@137 533 end.
adamc@137 534
adamc@137 535 (** This definition is accepted. It can be a little awkward to test Ltac definitions like this. Here is one method. *)
adamc@137 536
adamc@137 537 Goal False.
adamc@137 538 let n := length (1 :: 2 :: 3 :: nil) in
adamc@137 539 pose n.
adamc@137 540 (** [[
adamc@137 541 n := S (length (2 :: 3 :: nil)) : nat
adamc@137 542 ============================
adamc@137 543 False
adamc@220 544
adamc@137 545 ]]
adamc@137 546
adamc@220 547 We use the [pose] tactic, which extends the proof context with a new variable that is set equal to particular a term. We could also have used [idtac n] in place of [pose n], which would have printed the result without changing the context.
adamc@220 548
adamc@220 549 [n] only has the length calculation unrolled one step. What has happened here is that, by escaping into the [constr] nonterminal, we referred to the [length] function of Gallina, rather than the [length] Ltac function that we are defining. *)
adamc@220 550
adamc@220 551 Abort.
adamc@137 552
adamc@137 553 Reset length.
adamc@137 554
adamc@137 555 (** The thing to remember is that Gallina terms built by tactics must be bound explicitly via [let] or a similar technique, rather than inserting Ltac calls directly in other Gallina terms. *)
adamc@137 556
adamc@137 557 Ltac length ls :=
adamc@137 558 match ls with
adamc@137 559 | nil => O
adamc@137 560 | _ :: ?ls' =>
adamc@137 561 let ls'' := length ls' in
adamc@137 562 constr:(S ls'')
adamc@137 563 end.
adamc@137 564
adamc@137 565 Goal False.
adamc@137 566 let n := length (1 :: 2 :: 3 :: nil) in
adamc@137 567 pose n.
adamc@137 568 (** [[
adamc@137 569 n := 3 : nat
adamc@137 570 ============================
adamc@137 571 False
adamc@137 572 ]] *)
adamc@220 573
adamc@137 574 Abort.
adamc@141 575 (* end thide *)
adamc@141 576
adamc@141 577 (* EX: Write a list map function in Ltac. *)
adamc@137 578
adamc@137 579 (** We can also use anonymous function expressions and local function definitions in Ltac, as this example of a standard list [map] function shows. *)
adamc@137 580
adamc@141 581 (* begin thide *)
adamc@137 582 Ltac map T f :=
adamc@137 583 let rec map' ls :=
adamc@137 584 match ls with
adamc@137 585 | nil => constr:(@nil T)
adamc@137 586 | ?x :: ?ls' =>
adamc@137 587 let x' := f x in
adamc@137 588 let ls'' := map' ls' in
adamc@137 589 constr:(x' :: ls'')
adamc@137 590 end in
adamc@137 591 map'.
adamc@137 592
adamc@137 593 (** Ltac functions can have no implicit arguments. It may seem surprising that we need to pass [T], the carried type of the output list, explicitly. We cannot just use [type of f], because [f] is an Ltac term, not a Gallina term, and Ltac programs are dynamically typed. [f] could use very syntactic methods to decide to return differently typed terms for different inputs. We also could not replace [constr:(@nil T)] with [constr:nil], because we have no strongly-typed context to use to infer the parameter to [nil]. Luckily, we do have sufficient context within [constr:(x' :: ls'')].
adamc@137 594
adamc@137 595 Sometimes we need to employ the opposite direction of "nonterminal escape," when we want to pass a complicated tactic expression as an argument to another tactic, as we might want to do in invoking [map]. *)
adamc@137 596
adamc@137 597 Goal False.
adamc@137 598 let ls := map (nat * nat)%type ltac:(fun x => constr:(x, x)) (1 :: 2 :: 3 :: nil) in
adamc@137 599 pose ls.
adamc@137 600 (** [[
adamc@137 601 l := (1, 1) :: (2, 2) :: (3, 3) :: nil : list (nat * nat)
adamc@137 602 ============================
adamc@137 603 False
adamc@137 604 ]] *)
adamc@220 605
adamc@137 606 Abort.
adamc@141 607 (* end thide *)
adamc@137 608
adamc@138 609
adamc@139 610 (** * Recursive Proof Search *)
adamc@139 611
adamc@139 612 (** Deciding how to instantiate quantifiers is one of the hardest parts of automated first-order theorem proving. For a given problem, we can consider all possible bounded-length sequences of quantifier instantiations, applying only propositional reasoning at the end. This is probably a bad idea for almost all goals, but it makes for a nice example of recursive proof search procedures in Ltac.
adamc@139 613
adamc@139 614 We can consider the maximum "dependency chain" length for a first-order proof. We define the chain length for a hypothesis to be 0, and the chain length for an instantiation of a quantified fact to be one greater than the length for that fact. The tactic [inster n] is meant to try all possible proofs with chain length at most [n]. *)
adamc@139 615
adamc@141 616 (* begin thide *)
adamc@139 617 Ltac inster n :=
adamc@139 618 intuition;
adamc@139 619 match n with
adamc@139 620 | S ?n' =>
adamc@139 621 match goal with
adamc@139 622 | [ H : forall x : ?T, _, x : ?T |- _ ] => generalize (H x); inster n'
adamc@139 623 end
adamc@139 624 end.
adamc@141 625 (* end thide *)
adamc@139 626
adamc@139 627 (** [inster] begins by applying propositional simplification. Next, it checks if any chain length remains. If so, it tries all possible ways of instantiating quantified hypotheses with properly-typed local variables. It is critical to realize that, if the recursive call [inster n'] fails, then the [match goal] just seeks out another way of unifying its pattern against proof state. Thus, this small amount of code provides an elegant demonstration of how backtracking [match] enables exhaustive search.
adamc@139 628
adamc@139 629 We can verify the efficacy of [inster] with two short examples. The built-in [firstorder] tactic (with no extra arguments) is able to prove the first but not the second. *)
adamc@139 630
adamc@139 631 Section test_inster.
adamc@139 632 Variable A : Set.
adamc@139 633 Variables P Q : A -> Prop.
adamc@139 634 Variable f : A -> A.
adamc@139 635 Variable g : A -> A -> A.
adamc@139 636
adamc@139 637 Hypothesis H1 : forall x y, P (g x y) -> Q (f x).
adamc@139 638
adamc@139 639 Theorem test_inster : forall x y, P (g x y) -> Q (f x).
adamc@220 640 inster 2.
adamc@139 641 Qed.
adamc@139 642
adamc@139 643 Hypothesis H3 : forall u v, P u /\ P v /\ u <> v -> P (g u v).
adamc@139 644 Hypothesis H4 : forall u, Q (f u) -> P u /\ P (f u).
adamc@139 645
adamc@139 646 Theorem test_inster2 : forall x y, x <> y -> P x -> Q (f y) -> Q (f x).
adamc@220 647 inster 3.
adamc@139 648 Qed.
adamc@139 649 End test_inster.
adamc@139 650
adamc@140 651 (** The style employed in the definition of [inster] can seem very counterintuitive to functional programmers. Usually, functional programs accumulate state changes in explicit arguments to recursive functions. In Ltac, the state of the current subgoal is always implicit. Nonetheless, in contrast to general imperative programming, it is easy to undo any changes to this state, and indeed such "undoing" happens automatically at failures within [match]es. In this way, Ltac programming is similar to programming in Haskell with a stateful failure monad that supports a composition operator along the lines of the [first] tactical.
adamc@140 652
adamc@140 653 Functional programming purists may react indignantly to the suggestion of programming this way. Nonetheless, as with other kinds of "monadic programming," many problems are much simpler to solve with Ltac than they would be with explicit, pure proof manipulation in ML or Haskell. To demonstrate, we will write a basic simplification procedure for logical implications.
adamc@140 654
adamc@140 655 This procedure is inspired by one for separation logic, where conjuncts in formulas are thought of as "resources," such that we lose no completeness by "crossing out" equal conjuncts on the two sides of an implication. This process is complicated by the fact that, for reasons of modularity, our formulas can have arbitrary nested tree structure (branching at conjunctions) and may include existential quantifiers. It is helpful for the matching process to "go under" quantifiers and in fact decide how to instantiate existential quantifiers in the conclusion.
adamc@140 656
adamc@140 657 To distinguish the implications that our tactic handles from the implications that will show up as "plumbing" in various lemmas, we define a wrapper definition, a notation, and a tactic. *)
adamc@138 658
adamc@138 659 Definition imp (P1 P2 : Prop) := P1 -> P2.
adamc@140 660 Infix "-->" := imp (no associativity, at level 95).
adamc@140 661 Ltac imp := unfold imp; firstorder.
adamc@138 662
adamc@140 663 (** These lemmas about [imp] will be useful in the tactic that we will write. *)
adamc@138 664
adamc@138 665 Theorem and_True_prem : forall P Q,
adamc@138 666 (P /\ True --> Q)
adamc@138 667 -> (P --> Q).
adamc@138 668 imp.
adamc@138 669 Qed.
adamc@138 670
adamc@138 671 Theorem and_True_conc : forall P Q,
adamc@138 672 (P --> Q /\ True)
adamc@138 673 -> (P --> Q).
adamc@138 674 imp.
adamc@138 675 Qed.
adamc@138 676
adamc@138 677 Theorem assoc_prem1 : forall P Q R S,
adamc@138 678 (P /\ (Q /\ R) --> S)
adamc@138 679 -> ((P /\ Q) /\ R --> S).
adamc@138 680 imp.
adamc@138 681 Qed.
adamc@138 682
adamc@138 683 Theorem assoc_prem2 : forall P Q R S,
adamc@138 684 (Q /\ (P /\ R) --> S)
adamc@138 685 -> ((P /\ Q) /\ R --> S).
adamc@138 686 imp.
adamc@138 687 Qed.
adamc@138 688
adamc@138 689 Theorem comm_prem : forall P Q R,
adamc@138 690 (P /\ Q --> R)
adamc@138 691 -> (Q /\ P --> R).
adamc@138 692 imp.
adamc@138 693 Qed.
adamc@138 694
adamc@138 695 Theorem assoc_conc1 : forall P Q R S,
adamc@138 696 (S --> P /\ (Q /\ R))
adamc@138 697 -> (S --> (P /\ Q) /\ R).
adamc@138 698 imp.
adamc@138 699 Qed.
adamc@138 700
adamc@138 701 Theorem assoc_conc2 : forall P Q R S,
adamc@138 702 (S --> Q /\ (P /\ R))
adamc@138 703 -> (S --> (P /\ Q) /\ R).
adamc@138 704 imp.
adamc@138 705 Qed.
adamc@138 706
adamc@138 707 Theorem comm_conc : forall P Q R,
adamc@138 708 (R --> P /\ Q)
adamc@138 709 -> (R --> Q /\ P).
adamc@138 710 imp.
adamc@138 711 Qed.
adamc@138 712
adamc@140 713 (** The first order of business in crafting our [matcher] tactic will be auxiliary support for searching through formula trees. The [search_prem] tactic implements running its tactic argument [tac] on every subformula of an [imp] premise. As it traverses a tree, [search_prem] applies some of the above lemmas to rewrite the goal to bring different subformulas to the head of the goal. That is, for every subformula [P] of the implication premise, we want [P] to "have a turn," where the premise is rearranged into the form [P /\ Q] for some [Q]. The tactic [tac] should expect to see a goal in this form and focus its attention on the first conjunct of the premise. *)
adamc@140 714
adamc@141 715 (* begin thide *)
adamc@138 716 Ltac search_prem tac :=
adamc@138 717 let rec search P :=
adamc@138 718 tac
adamc@138 719 || (apply and_True_prem; tac)
adamc@138 720 || match P with
adamc@138 721 | ?P1 /\ ?P2 =>
adamc@138 722 (apply assoc_prem1; search P1)
adamc@138 723 || (apply assoc_prem2; search P2)
adamc@138 724 end
adamc@138 725 in match goal with
adamc@138 726 | [ |- ?P /\ _ --> _ ] => search P
adamc@138 727 | [ |- _ /\ ?P --> _ ] => apply comm_prem; search P
adamc@138 728 | [ |- _ --> _ ] => progress (tac || (apply and_True_prem; tac))
adamc@138 729 end.
adamc@138 730
adamc@140 731 (** To understand how [search_prem] works, we turn first to the final [match]. If the premise begins with a conjunction, we call the [search] procedure on each of the conjuncts, or only the first conjunct, if that already yields a case where [tac] does not fail. [search P] expects and maintains the invariant that the premise is of the form [P /\ Q] for some [Q]. We pass [P] explicitly as a kind of decreasing induction measure, to avoid looping forever when [tac] always fails. The second [match] case calls a commutativity lemma to realize this invariant, before passing control to [search]. The final [match] case tries applying [tac] directly and then, if that fails, changes the form of the goal by adding an extraneous [True] conjunct and calls [tac] again.
adamc@140 732
adamc@140 733 [search] itself tries the same tricks as in the last case of the final [match]. Additionally, if neither works, it checks if [P] is a conjunction. If so, it calls itself recursively on each conjunct, first applying associativity lemmas to maintain the goal-form invariant.
adamc@140 734
adamc@140 735 We will also want a dual function [search_conc], which does tree search through an [imp] conclusion. *)
adamc@140 736
adamc@138 737 Ltac search_conc tac :=
adamc@138 738 let rec search P :=
adamc@138 739 tac
adamc@138 740 || (apply and_True_conc; tac)
adamc@138 741 || match P with
adamc@138 742 | ?P1 /\ ?P2 =>
adamc@138 743 (apply assoc_conc1; search P1)
adamc@138 744 || (apply assoc_conc2; search P2)
adamc@138 745 end
adamc@138 746 in match goal with
adamc@138 747 | [ |- _ --> ?P /\ _ ] => search P
adamc@138 748 | [ |- _ --> _ /\ ?P ] => apply comm_conc; search P
adamc@138 749 | [ |- _ --> _ ] => progress (tac || (apply and_True_conc; tac))
adamc@138 750 end.
adamc@138 751
adamc@140 752 (** Now we can prove a number of lemmas that are suitable for application by our search tactics. A lemma that is meant to handle a premise should have the form [P /\ Q --> R] for some interesting [P], and a lemma that is meant to handle a conclusion should have the form [P --> Q /\ R] for some interesting [Q]. *)
adamc@140 753
adamc@138 754 Theorem False_prem : forall P Q,
adamc@138 755 False /\ P --> Q.
adamc@138 756 imp.
adamc@138 757 Qed.
adamc@138 758
adamc@138 759 Theorem True_conc : forall P Q : Prop,
adamc@138 760 (P --> Q)
adamc@138 761 -> (P --> True /\ Q).
adamc@138 762 imp.
adamc@138 763 Qed.
adamc@138 764
adamc@138 765 Theorem Match : forall P Q R : Prop,
adamc@138 766 (Q --> R)
adamc@138 767 -> (P /\ Q --> P /\ R).
adamc@138 768 imp.
adamc@138 769 Qed.
adamc@138 770
adamc@138 771 Theorem ex_prem : forall (T : Type) (P : T -> Prop) (Q R : Prop),
adamc@138 772 (forall x, P x /\ Q --> R)
adamc@138 773 -> (ex P /\ Q --> R).
adamc@138 774 imp.
adamc@138 775 Qed.
adamc@138 776
adamc@138 777 Theorem ex_conc : forall (T : Type) (P : T -> Prop) (Q R : Prop) x,
adamc@138 778 (Q --> P x /\ R)
adamc@138 779 -> (Q --> ex P /\ R).
adamc@138 780 imp.
adamc@138 781 Qed.
adamc@138 782
adamc@140 783 (** We will also want a "base case" lemma for finishing proofs where cancelation has removed every constituent of the conclusion. *)
adamc@140 784
adamc@138 785 Theorem imp_True : forall P,
adamc@138 786 P --> True.
adamc@138 787 imp.
adamc@138 788 Qed.
adamc@138 789
adamc@220 790 (** Our final [matcher] tactic is now straightforward. First, we [intros] all variables into scope. Then we attempt simple premise simplifications, finishing the proof upon finding [False] and eliminating any existential quantifiers that we find. After that, we search through the conclusion. We remove [True] conjuncts, remove existential quantifiers by introducing unification variables for their bound variables, and search for matching premises to cancel. Finally, when no more progress is made, we see if the goal has become trivial and can be solved by [imp_True]. In each case, we use the tactic [simple apply] in place of [apply] to use a simpler, less expensive unification algorithm. *)
adamc@140 791
adamc@138 792 Ltac matcher :=
adamc@138 793 intros;
adamc@204 794 repeat search_prem ltac:(simple apply False_prem || (simple apply ex_prem; intro));
adamc@204 795 repeat search_conc ltac:(simple apply True_conc || simple eapply ex_conc
adamc@204 796 || search_prem ltac:(simple apply Match));
adamc@204 797 try simple apply imp_True.
adamc@141 798 (* end thide *)
adamc@140 799
adamc@140 800 (** Our tactic succeeds at proving a simple example. *)
adamc@138 801
adamc@138 802 Theorem t2 : forall P Q : Prop,
adamc@138 803 Q /\ (P /\ False) /\ P --> P /\ Q.
adamc@138 804 matcher.
adamc@138 805 Qed.
adamc@138 806
adamc@140 807 (** In the generated proof, we find a trace of the workings of the search tactics. *)
adamc@140 808
adamc@140 809 Print t2.
adamc@220 810 (** %\vspace{-.15in}% [[
adamc@140 811 t2 =
adamc@140 812 fun P Q : Prop =>
adamc@140 813 comm_prem (assoc_prem1 (assoc_prem2 (False_prem (P:=P /\ P /\ Q) (P /\ Q))))
adamc@140 814 : forall P Q : Prop, Q /\ (P /\ False) /\ P --> P /\ Q
adamc@220 815
adamc@220 816 ]]
adamc@140 817
adamc@220 818 We can also see that [matcher] is well-suited for cases where some human intervention is needed after the automation finishes. *)
adamc@140 819
adamc@138 820 Theorem t3 : forall P Q R : Prop,
adamc@138 821 P /\ Q --> Q /\ R /\ P.
adamc@138 822 matcher.
adamc@140 823 (** [[
adamc@140 824 ============================
adamc@140 825 True --> R
adamc@220 826
adamc@140 827 ]]
adamc@140 828
adamc@140 829 [matcher] canceled those conjuncts that it was able to cancel, leaving a simplified subgoal for us, much as [intuition] does. *)
adamc@220 830
adamc@138 831 Abort.
adamc@138 832
adamc@140 833 (** [matcher] even succeeds at guessing quantifier instantiations. It is the unification that occurs in uses of the [Match] lemma that does the real work here. *)
adamc@140 834
adamc@138 835 Theorem t4 : forall (P : nat -> Prop) Q, (exists x, P x /\ Q) --> Q /\ (exists x, P x).
adamc@138 836 matcher.
adamc@138 837 Qed.
adamc@138 838
adamc@140 839 Print t4.
adamc@220 840 (** %\vspace{-.15in}% [[
adamc@140 841 t4 =
adamc@140 842 fun (P : nat -> Prop) (Q : Prop) =>
adamc@140 843 and_True_prem
adamc@140 844 (ex_prem (P:=fun x : nat => P x /\ Q)
adamc@140 845 (fun x : nat =>
adamc@140 846 assoc_prem2
adamc@140 847 (Match (P:=Q)
adamc@140 848 (and_True_conc
adamc@140 849 (ex_conc (fun x0 : nat => P x0) x
adamc@140 850 (Match (P:=P x) (imp_True (P:=True))))))))
adamc@140 851 : forall (P : nat -> Prop) (Q : Prop),
adamc@140 852 (exists x : nat, P x /\ Q) --> Q /\ (exists x : nat, P x)
adamc@140 853 ]] *)
adamc@234 854
adamc@234 855
adamc@234 856 (** * Creating Unification Variables *)
adamc@234 857
adamc@234 858 (** A final useful ingredient in tactic crafting is the ability to allocate new unification variables explicitly. Tactics like [eauto] introduce unification variable internally to support flexible proof search. While [eauto] and its relatives do %\textit{%#<i>#backward#</i>#%}% reasoning, we often want to do similar %\textit{%#<i>#forward#</i>#%}% reasoning, where unification variables can be useful for similar reasons.
adamc@234 859
adamc@234 860 For example, we can write a tactic that instantiates the quantifiers of a universally-quantified hypothesis. The tactic should not need to know what the appropriate instantiantiations are; rather, we want these choices filled with placeholders. We hope that, when we apply the specialized hypothesis later, syntactic unification will determine concrete values.
adamc@234 861
adamc@234 862 Before we are ready to write a tactic, we can try out its ingredients one at a time. *)
adamc@234 863
adamc@234 864 Theorem t5 : (forall x : nat, S x > x) -> 2 > 1.
adamc@234 865 intros.
adamc@234 866
adamc@234 867 (** [[
adamc@234 868 H : forall x : nat, S x > x
adamc@234 869 ============================
adamc@234 870 2 > 1
adamc@234 871
adamc@234 872 ]]
adamc@234 873
adamc@234 874 To instantiate [H] generically, we first need to name the value to be used for [x]. *)
adamc@234 875
adamc@234 876 evar (y : nat).
adamc@234 877
adamc@234 878 (** [[
adamc@234 879 H : forall x : nat, S x > x
adamc@234 880 y := ?279 : nat
adamc@234 881 ============================
adamc@234 882 2 > 1
adamc@234 883
adamc@234 884 ]]
adamc@234 885
adamc@234 886 The proof context is extended with a new variable [y], which has been assigned to be equal to a fresh unification variable [?279]. We want to instantiate [H] with [?279]. To get ahold of the new unification variable, rather than just its alias [y], we perform a trivial call-by-value reduction in the expression [y]. In particular, we only request the use of one reduction rule, [delta], which deals with definition unfolding. We pass a flag further stipulating that only the definition of [y] be unfolded. This is a simple trick for getting at the value of a synonym variable. *)
adamc@234 887
adamc@234 888 let y' := eval cbv delta [y] in y in
adamc@234 889 clear y; generalize (H y').
adamc@234 890
adamc@234 891 (** [[
adamc@234 892 H : forall x : nat, S x > x
adamc@234 893 ============================
adamc@234 894 S ?279 > ?279 -> 2 > 1
adamc@234 895
adamc@234 896 ]]
adamc@234 897
adamc@234 898 Our instantiation was successful. We can finish by using the refined formula to replace the original. *)
adamc@234 899
adamc@234 900 clear H; intro H.
adamc@234 901
adamc@234 902 (** [[
adamc@234 903 H : S ?281 > ?281
adamc@234 904 ============================
adamc@234 905 2 > 1
adamc@234 906
adamc@234 907 ]]
adamc@234 908
adamc@234 909 We can finish the proof by using [apply]'s unification to figure out the proper value of [?281]. (The original unification variable was replaced by another, as often happens in the internals of the various tactics' implementations.) *)
adamc@234 910
adamc@234 911 apply H.
adamc@234 912 Qed.
adamc@234 913
adamc@234 914 (** Now we can write a tactic that encapsulates the pattern we just employed, instantiating all quantifiers of a particular hypothesis. *)
adamc@234 915
adamc@234 916 Ltac insterU H :=
adamc@234 917 repeat match type of H with
adamc@234 918 | forall x : ?T, _ =>
adamc@234 919 let x := fresh "x" in
adamc@234 920 evar (x : T);
adamc@234 921 let x' := eval cbv delta [x] in x in
adamc@234 922 clear x; generalize (H x'); clear H; intro H
adamc@234 923 end.
adamc@234 924
adamc@234 925 Theorem t5' : (forall x : nat, S x > x) -> 2 > 1.
adamc@234 926 intro H; insterU H; apply H.
adamc@234 927 Qed.
adamc@234 928
adamc@234 929 (** This particular example is somewhat silly, since [apply] by itself would have solved the goal originally. Separate forward reasoning is more useful on hypotheses that end in existential quantifications. Before we go through an example, it is useful to define a variant of [insterU] that does not clear the base hypothesis we pass to it. *)
adamc@234 930
adamc@234 931 Ltac insterKeep H :=
adamc@234 932 let H' := fresh "H'" in
adamc@234 933 generalize H; intro H'; insterU H'.
adamc@234 934
adamc@234 935 Section t6.
adamc@234 936 Variables A B : Type.
adamc@234 937 Variable P : A -> B -> Prop.
adamc@234 938 Variable f : A -> A -> A.
adamc@234 939 Variable g : B -> B -> B.
adamc@234 940
adamc@234 941 Hypothesis H1 : forall v, exists u, P v u.
adamc@234 942 Hypothesis H2 : forall v1 u1 v2 u2,
adamc@234 943 P v1 u1
adamc@234 944 -> P v2 u2
adamc@234 945 -> P (f v1 v2) (g u1 u2).
adamc@234 946
adamc@234 947 Theorem t6 : forall v1 v2, exists u1, exists u2, P (f v1 v2) (g u1 u2).
adamc@234 948 intros.
adamc@234 949
adamc@234 950 (** Neither [eauto] nor [firstorder] is clever enough to prove this goal. We can help out by doing some of the work with quantifiers ourselves. *)
adamc@234 951
adamc@234 952 do 2 insterKeep H1.
adamc@234 953
adamc@234 954 (** Our proof state is extended with two generic instances of [H1].
adamc@234 955
adamc@234 956 [[
adamc@234 957 H' : exists u : B, P ?4289 u
adamc@234 958 H'0 : exists u : B, P ?4288 u
adamc@234 959 ============================
adamc@234 960 exists u1 : B, exists u2 : B, P (f v1 v2) (g u1 u2)
adamc@234 961
adamc@234 962 ]]
adamc@234 963
adamc@234 964 [eauto] still cannot prove the goal, so we eliminate the two new existential quantifiers. *)
adamc@234 965
adamc@234 966 repeat match goal with
adamc@234 967 | [ H : ex _ |- _ ] => destruct H
adamc@234 968 end.
adamc@234 969
adamc@234 970 (** Now the goal is simple enough to solve by logic programming. *)
adamc@234 971
adamc@234 972 eauto.
adamc@234 973 Qed.
adamc@234 974 End t6.
adamc@234 975
adamc@234 976 (** Our [insterU] tactic does not fare so well with quantified hypotheses that also contain implications. We can see the problem in a slight modification of the last example. We introduce a new unary predicate [Q] and use it to state an additional requirement of our hypothesis [H1]. *)
adamc@234 977
adamc@234 978 Section t7.
adamc@234 979 Variables A B : Type.
adamc@234 980 Variable Q : A -> Prop.
adamc@234 981 Variable P : A -> B -> Prop.
adamc@234 982 Variable f : A -> A -> A.
adamc@234 983 Variable g : B -> B -> B.
adamc@234 984
adamc@234 985 Hypothesis H1 : forall v, Q v -> exists u, P v u.
adamc@234 986 Hypothesis H2 : forall v1 u1 v2 u2,
adamc@234 987 P v1 u1
adamc@234 988 -> P v2 u2
adamc@234 989 -> P (f v1 v2) (g u1 u2).
adamc@234 990
adamc@234 991 Theorem t6 : forall v1 v2, Q v1 -> Q v2 -> exists u1, exists u2, P (f v1 v2) (g u1 u2).
adamc@234 992 intros; do 2 insterKeep H1;
adamc@234 993 repeat match goal with
adamc@234 994 | [ H : ex _ |- _ ] => destruct H
adamc@234 995 end; eauto.
adamc@234 996
adamc@234 997 (** This proof script does not hit any errors until the very end, when an error message like this one is displayed.
adamc@234 998
adamc@234 999 [[
adamc@234 1000 No more subgoals but non-instantiated existential variables :
adamc@234 1001 Existential 1 =
adamc@234 1002 ?4384 : [A : Type
adamc@234 1003 B : Type
adamc@234 1004 Q : A -> Prop
adamc@234 1005 P : A -> B -> Prop
adamc@234 1006 f : A -> A -> A
adamc@234 1007 g : B -> B -> B
adamc@234 1008 H1 : forall v : A, Q v -> exists u : B, P v u
adamc@234 1009 H2 : forall (v1 : A) (u1 : B) (v2 : A) (u2 : B),
adamc@234 1010 P v1 u1 -> P v2 u2 -> P (f v1 v2) (g u1 u2)
adamc@234 1011 v1 : A
adamc@234 1012 v2 : A
adamc@234 1013 H : Q v1
adamc@234 1014 H0 : Q v2
adamc@234 1015 H' : Q v2 -> exists u : B, P v2 u |- Q v2]
adamc@234 1016
adamc@234 1017 ]]
adamc@234 1018
adamc@234 1019 There is another similar line about a different existential variable. Here, "existential variable" means what we have also called "unification variable." In the course of the proof, some unification variable [?4384] was introduced but never unified. Unification variables are just a device to structure proof search; the language of Gallina proof terms does not include them. Thus, we cannot produce a proof term without instantiating the variable.
adamc@234 1020
adamc@234 1021 The error message shows that [?4384] is meant to be a proof of [Q v2] in a particular proof state, whose variables and hypotheses are displayed. It turns out that [?4384] was created by [insterU], as the value of a proof to pass to [H1]. Recall that, in Gallina, implication is just a degenerate case of [forall] quantification, so the [insterU] code to match against [forall] also matched the implication. Since any proof of [Q v2] is as good as any other in this context, there was never any opportunity to use unification to determine exactly which proof is appropriate. We expect similar problems with any implications in arguments to [insterU]. *)
adamc@234 1022
adamc@234 1023 Abort.
adamc@234 1024 End t7.
adamc@234 1025
adamc@234 1026 Reset insterU.
adamc@234 1027
adamc@234 1028 (** We can redefine [insterU] to treat implications differently. In particular, we pattern-match on the type of the type [T] in [forall x : ?T, ...]. If [T] has type [Prop], then [x]'s instantiation should be thought of as a proof. Thus, instead of picking a new unification variable for it, we instead apply a user-supplied tactic [tac]. It is important that we end this special [Prop] case with [|| fail 1], so that, if [tac] fails to prove [T], we abort the instantiation, rather than continuing on to the default quantifier handling. *)
adamc@234 1029
adamc@234 1030 Ltac insterU tac H :=
adamc@234 1031 repeat match type of H with
adamc@234 1032 | forall x : ?T, _ =>
adamc@234 1033 match type of T with
adamc@234 1034 | Prop =>
adamc@234 1035 (let H' := fresh "H'" in
adamc@234 1036 assert (H' : T); [
adamc@234 1037 solve [ tac ]
adamc@234 1038 | generalize (H H'); clear H H'; intro H ])
adamc@234 1039 || fail 1
adamc@234 1040 | _ =>
adamc@234 1041 let x := fresh "x" in
adamc@234 1042 evar (x : T);
adamc@234 1043 let x' := eval cbv delta [x] in x in
adamc@234 1044 clear x; generalize (H x'); clear H; intro H
adamc@234 1045 end
adamc@234 1046 end.
adamc@234 1047
adamc@234 1048 Ltac insterKeep tac H :=
adamc@234 1049 let H' := fresh "H'" in
adamc@234 1050 generalize H; intro H'; insterU tac H'.
adamc@234 1051
adamc@234 1052 Section t7.
adamc@234 1053 Variables A B : Type.
adamc@234 1054 Variable Q : A -> Prop.
adamc@234 1055 Variable P : A -> B -> Prop.
adamc@234 1056 Variable f : A -> A -> A.
adamc@234 1057 Variable g : B -> B -> B.
adamc@234 1058
adamc@234 1059 Hypothesis H1 : forall v, Q v -> exists u, P v u.
adamc@234 1060 Hypothesis H2 : forall v1 u1 v2 u2,
adamc@234 1061 P v1 u1
adamc@234 1062 -> P v2 u2
adamc@234 1063 -> P (f v1 v2) (g u1 u2).
adamc@234 1064
adamc@234 1065 Theorem t6 : forall v1 v2, Q v1 -> Q v2 -> exists u1, exists u2, P (f v1 v2) (g u1 u2).
adamc@234 1066
adamc@234 1067 (** We can prove the goal by calling [insterKeep] with a tactic that tries to find and apply a [Q] hypothesis over a variable about which we do not yet know any [P] facts. We need to begin this tactic code with [idtac; ] to get around a strange limitation in Coq's proof engine, where a first-class tactic argument may not begin with a [match]. *)
adamc@234 1068
adamc@234 1069 intros; do 2 insterKeep ltac:(idtac; match goal with
adamc@234 1070 | [ H : Q ?v |- _ ] =>
adamc@234 1071 match goal with
adamc@234 1072 | [ _ : context[P v _] |- _ ] => fail 1
adamc@234 1073 | _ => apply H
adamc@234 1074 end
adamc@234 1075 end) H1;
adamc@234 1076 repeat match goal with
adamc@234 1077 | [ H : ex _ |- _ ] => destruct H
adamc@234 1078 end; eauto.
adamc@234 1079 Qed.
adamc@234 1080 End t7.
adamc@234 1081
adamc@234 1082 (** It is often useful to instantiate existential variables explicitly. A built-in tactic provides one way of doing so. *)
adamc@234 1083
adamc@234 1084 Theorem t8 : exists p : nat * nat, fst p = 3.
adamc@234 1085 econstructor; instantiate (1 := (3, 2)); reflexivity.
adamc@234 1086 Qed.
adamc@234 1087
adamc@234 1088 (** The [1] above is identifying an existential variable appearing in the current goal, with the last existential appearing assigned number 1, the second last assigned number 2, and so on. The named existential is replaced everywhere by the term to the right of the [:=].
adamc@234 1089
adamc@234 1090 The [instantiate] tactic can be convenient for exploratory proving, but it leads to very brittle proof scripts that are unlikely to adapt to changing theorem statements. It is often more helpful to have a tactic that can be used to assign a value to a term that is known to be an existential. By employing a roundabout implementation technique, we can build a tactic that generalizes this functionality. In particular, our tactic [equate] will assert that two terms are equal. If one of the terms happens to be an existential, then it will be replaced everywhere with the other term. *)
adamc@234 1091
adamc@234 1092 Ltac equate x y :=
adamc@234 1093 let H := fresh "H" in
adamc@234 1094 assert (H : x = y); [ reflexivity | clear H ].
adamc@234 1095
adamc@234 1096 (** [equate] fails if it is not possible to prove [x = y] by [reflexivity]. We perform the proof only for its unification side effects, clearing the fact [x = y] afterward. With [equate], we can build a less brittle version of the prior example. *)
adamc@234 1097
adamc@234 1098 Theorem t9 : exists p : nat * nat, fst p = 3.
adamc@234 1099 econstructor; match goal with
adamc@234 1100 | [ |- fst ?x = 3 ] => equate x (3, 2)
adamc@234 1101 end; reflexivity.
adamc@234 1102 Qed.