Certified Programming with Dependent Types: src/Match.v annotate

annotate src/Match.v @ 305:690796f4690d

Further emphasize necessity and purpose of Set Implicit Arguments; tweak Makefile to support parallel builds

author	Adam Chlipala <adam@chlipala.net>
date	Wed, 03 Aug 2011 10:34:53 -0400
parents	7b38729be069
children	d5787b70cf48

rev	line source
adam@297	1 (* Copyright (c) 2008-2011, 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
adam@288	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
adam@288	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
adam@288	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
adam@288	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
adam@302	132 ]]
adam@302	133 *)
adamc@220	134
adamc@134	135 Abort.
adamc@134	136
adam@288	137 (** 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. The key difference is that [auto] either solves a goal or makes no changes to it, while [autorewrite] may change goals without solving them. The situation for [eauto] is slightly more complicated, as changes to hint databases may change the proof found for a particular goal, and that proof may influence the settings of unification variables that appear elsewhere in the proof state. *)
adamc@134	138
adamc@134	139 Reset garden_path.
adamc@134	140
adam@288	141 (** [autorewrite] also works with quantified equalities that include additional premises, but we must be careful to avoid similar incorrect rewritings. *)
adamc@134	142
adamc@134	143 Section garden_path.
adamc@134	144 Variable P : A -> Prop.
adamc@134	145 Variable g : A -> A.
adamc@134	146 Hypothesis f_g : forall x, P x -> f x = g x.
adamc@134	147 Hint Rewrite f_g : my_db.
adamc@134	148
adamc@134	149 Lemma f_f_f' : forall x, f (f (f x)) = f x.
adamc@134	150 intros; autorewrite with my_db.
adamc@134	151 (** [[
adamc@134	152
adamc@134	153 ============================
adamc@134	154 g (g (g x)) = g x
adamc@134	155
adamc@134	156 subgoal 2 is:
adamc@134	157 P x
adamc@134	158 subgoal 3 is:
adamc@134	159 P (f x)
adamc@134	160 subgoal 4 is:
adamc@134	161 P (f x)
adam@302	162 ]]
adam@302	163 *)
adamc@220	164
adamc@134	165 Abort.
adamc@134	166
adamc@134	167 (** 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	168
adamc@134	169 Reset garden_path.
adamc@134	170
adam@288	171 (** Our final, successful, attempt uses an extra argument to [Hint Rewrite] that specifies a tactic to apply to generated premises. Such a hint is only used when the tactic succeeds for all premises, possibly leaving further subgoals for some premises. *)
adamc@134	172
adamc@134	173 Section garden_path.
adamc@134	174 Variable P : A -> Prop.
adamc@134	175 Variable g : A -> A.
adamc@134	176 Hypothesis f_g : forall x, P x -> f x = g x.
adamc@141	177 (* begin thide *)
adamc@134	178 Hint Rewrite f_g using assumption : my_db.
adamc@141	179 (* end thide *)
adamc@134	180
adamc@134	181 Lemma f_f_f' : forall x, f (f (f x)) = f x.
adamc@141	182 (* begin thide *)
adamc@134	183 intros; autorewrite with my_db; reflexivity.
adamc@134	184 Qed.
adamc@141	185 (* end thide *)
adamc@134	186
adamc@134	187 (** [autorewrite] will still use [f_g] when the generated premise is among our assumptions. *)
adamc@134	188
adamc@134	189 Lemma f_f_f_g : forall x, P x -> f (f x) = g x.
adamc@141	190 (* begin thide *)
adamc@134	191 intros; autorewrite with my_db; reflexivity.
adamc@141	192 (* end thide *)
adamc@134	193 Qed.
adamc@134	194 End garden_path.
adamc@134	195
adamc@220	196 (** remove printing * *)
adamc@220	197
adamc@134	198 (** 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	199
adamc@220	200 (** printing * $$ )
adamc@220	201
adamc@134	202 Lemma in_star : forall x y, f (f (f (f x))) = f (f y)
adamc@134	203 -> f x = f (f (f y)).
adamc@141	204 (* begin thide *)
adamc@134	205 intros; autorewrite with my_db in *; assumption.
adamc@141	206 (* end thide *)
adamc@134	207 Qed.
adamc@134	208
adamc@134	209 End autorewrite.
adamc@135	210
adamc@135	211
adamc@135	212 (** * Ltac Programming Basics *)
adamc@135	213
adamc@135	214 (** 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	215
adamc@135	216 One common use for [match] tactics is identification of subjects for case analysis, as we see in this tactic definition. *)
adamc@135	217
adamc@141	218 (* begin thide *)
adamc@135	219 Ltac find_if :=
adamc@135	220 match goal with
adamc@135	221 \| [ \|- if ?X then _ else _ ] => destruct X
adamc@135	222 end.
adamc@141	223 (* end thide *)
adamc@135	224
adamc@135	225 (** 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	226
adamc@135	227 Theorem hmm : forall (a b c : bool),
adamc@135	228 if a
adamc@135	229 then if b
adamc@135	230 then True
adamc@135	231 else True
adamc@135	232 else if c
adamc@135	233 then True
adamc@135	234 else True.
adamc@141	235 (* begin thide *)
adamc@135	236 intros; repeat find_if; constructor.
adamc@135	237 Qed.
adamc@141	238 (* end thide *)
adamc@135	239
adamc@135	240 (** 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	241
adamc@135	242 Another very useful Ltac building block is %\textit{%#<i>#context patterns#</i>#%}%. *)
adamc@135	243
adamc@141	244 (* begin thide *)
adamc@135	245 Ltac find_if_inside :=
adamc@135	246 match goal with
adamc@135	247 \| [ \|- context[if ?X then _ else _] ] => destruct X
adamc@135	248 end.
adamc@141	249 (* end thide *)
adamc@135	250
adamc@135	251 (** 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	252
adamc@135	253 Theorem hmm' : forall (a b c : bool),
adamc@135	254 if a
adamc@135	255 then if b
adamc@135	256 then True
adamc@135	257 else True
adamc@135	258 else if c
adamc@135	259 then True
adamc@135	260 else True.
adamc@141	261 (* begin thide *)
adamc@135	262 intros; repeat find_if_inside; constructor.
adamc@135	263 Qed.
adamc@141	264 (* end thide *)
adamc@135	265
adamc@135	266 (** We can also use [find_if_inside] to prove goals that [find_if] does not simplify sufficiently. *)
adamc@135	267
adamc@141	268 Theorem hmm2 : forall (a b : bool),
adamc@135	269 (if a then 42 else 42) = (if b then 42 else 42).
adamc@141	270 (* begin thide *)
adamc@135	271 intros; repeat find_if_inside; reflexivity.
adamc@135	272 Qed.
adamc@141	273 (* end thide *)
adamc@135	274
adam@288	275 (** 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	276
adamc@141	277 (* begin thide *)
adamc@135	278 Ltac my_tauto :=
adamc@135	279 repeat match goal with
adamc@135	280 \| [ H : ?P \|- ?P ] => exact H
adamc@135	281
adamc@135	282 \| [ \|- True ] => constructor
adamc@135	283 \| [ \|- _ /\ _ ] => constructor
adamc@135	284 \| [ \|- _ -> _ ] => intro
adamc@135	285
adamc@135	286 \| [ H : False \|- _ ] => destruct H
adamc@135	287 \| [ H : _ /\ _ \|- _ ] => destruct H
adamc@135	288 \| [ H : _ \/ _ \|- _ ] => destruct H
adamc@135	289
adamc@135	290 \| [ H1 : ?P -> ?Q, H2 : ?P \|- _ ] =>
adamc@135	291 let H := fresh "H" in
adamc@135	292 generalize (H1 H2); clear H1; intro H
adamc@135	293 end.
adamc@141	294 (* end thide *)
adamc@135	295
adamc@135	296 (** 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	297
adam@288	298 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	299
adamc@135	300 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	301
adamc@135	302 Section propositional.
adamc@135	303 Variables P Q R : Prop.
adamc@135	304
adamc@138	305 Theorem propositional : (P \/ Q \/ False) /\ (P -> Q) -> True /\ Q.
adamc@141	306 (* begin thide *)
adamc@135	307 my_tauto.
adamc@135	308 Qed.
adamc@141	309 (* end thide *)
adamc@135	310 End propositional.
adamc@135	311
adamc@135	312 (** 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	313
adamc@135	314 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	315
adamc@135	316 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	317
adamc@135	318 For instance, this (unnecessarily verbose) proof script works: *)
adamc@135	319
adamc@135	320 Theorem m1 : True.
adamc@135	321 match goal with
adamc@135	322 \| [ \|- _ ] => intro
adamc@135	323 \| [ \|- True ] => constructor
adamc@135	324 end.
adamc@141	325 (* begin thide *)
adamc@135	326 Qed.
adamc@141	327 (* end thide *)
adamc@135	328
adamc@135	329 (** 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	330
adamc@135	331 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	332
adamc@135	333 Theorem m2 : forall P Q R : Prop, P -> Q -> R -> Q.
adamc@135	334 intros; match goal with
adamc@220	335 \| [ H : _ \|- _ ] => idtac H
adamc@135	336 end.
adamc@135	337
adam@288	338 (** 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	339
adamc@141	340 (* begin thide *)
adamc@135	341 match goal with
adamc@135	342 \| [ H : _ \|- _ ] => exact H
adamc@135	343 end.
adamc@135	344 Qed.
adamc@141	345 (* end thide *)
adamc@135	346
adamc@135	347 (** 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	348
adamc@135	349 (** Now we are equipped to implement a tactic for checking that a proposition is not among our hypotheses: *)
adamc@135	350
adamc@141	351 (* begin thide *)
adamc@135	352 Ltac notHyp P :=
adamc@135	353 match goal with
adamc@135	354 \| [ _ : P \|- _ ] => fail 1
adamc@135	355 \| _ =>
adamc@135	356 match P with
adamc@135	357 \| ?P1 /\ ?P2 => first [ notHyp P1 \| notHyp P2 \| fail 2 ]
adamc@135	358 \| _ => idtac
adamc@135	359 end
adamc@135	360 end.
adamc@141	361 (* end thide *)
adamc@135	362
adam@288	363 (** 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	364
adamc@135	365 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	366
adamc@135	367 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	368
adamc@135	369 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	370
adamc@141	371 (* begin thide *)
adamc@135	372 Ltac extend pf :=
adamc@135	373 let t := type of pf in
adamc@135	374 notHyp t; generalize pf; intro.
adamc@141	375 (* end thide *)
adamc@135	376
adamc@135	377 (** 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	378
adamc@135	379 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	380
adamc@141	381 (* begin thide *)
adamc@135	382 Ltac completer :=
adamc@135	383 repeat match goal with
adamc@135	384 \| [ \|- _ /\ _ ] => constructor
adamc@135	385 \| [ H : _ /\ _ \|- _ ] => destruct H
adamc@135	386 \| [ H : ?P -> ?Q, H' : ?P \|- _ ] =>
adamc@135	387 generalize (H H'); clear H; intro H
adamc@135	388 \| [ \|- forall x, _ ] => intro
adamc@135	389
adamc@135	390 \| [ H : forall x, ?P x -> _, H' : ?P ?X \|- _ ] =>
adamc@135	391 extend (H X H')
adamc@135	392 end.
adamc@141	393 (* end thide *)
adamc@135	394
adamc@135	395 (** 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	396
adamc@135	397 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	398
adamc@135	399 We can check that [completer] is working properly: *)
adamc@135	400
adamc@135	401 Section firstorder.
adamc@135	402 Variable A : Set.
adamc@135	403 Variables P Q R S : A -> Prop.
adamc@135	404
adamc@135	405 Hypothesis H1 : forall x, P x -> Q x /\ R x.
adamc@135	406 Hypothesis H2 : forall x, R x -> S x.
adamc@135	407
adamc@135	408 Theorem fo : forall x, P x -> S x.
adamc@141	409 (* begin thide *)
adamc@135	410 completer.
adamc@135	411 (** [[
adamc@135	412 x : A
adamc@135	413 H : P x
adamc@135	414 H0 : Q x
adamc@135	415 H3 : R x
adamc@135	416 H4 : S x
adamc@135	417 ============================
adamc@135	418 S x
adam@302	419 ]]
adam@302	420 *)
adamc@135	421
adamc@135	422 assumption.
adamc@135	423 Qed.
adamc@141	424 (* end thide *)
adamc@135	425 End firstorder.
adamc@135	426
adamc@135	427 (** 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	428
adamc@141	429 (* begin thide *)
adamc@135	430 Ltac completer' :=
adamc@135	431 repeat match goal with
adamc@135	432 \| [ \|- _ /\ _ ] => constructor
adamc@135	433 \| [ H : _ /\ _ \|- _ ] => destruct H
adamc@135	434 \| [ H : ?P -> _, H' : ?P \|- _ ] =>
adamc@135	435 generalize (H H'); clear H; intro H
adamc@135	436 \| [ \|- forall x, _ ] => intro
adamc@135	437
adamc@135	438 \| [ H : forall x, ?P x -> _, H' : ?P ?X \|- _ ] =>
adamc@135	439 extend (H X H')
adamc@135	440 end.
adamc@141	441 (* end thide *)
adamc@135	442
adamc@135	443 (** 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	444
adamc@135	445 Section firstorder'.
adamc@135	446 Variable A : Set.
adamc@135	447 Variables P Q R S : A -> Prop.
adamc@135	448
adamc@135	449 Hypothesis H1 : forall x, P x -> Q x /\ R x.
adamc@135	450 Hypothesis H2 : forall x, R x -> S x.
adamc@135	451
adamc@135	452 Theorem fo' : forall x, P x -> S x.
adamc@141	453 (* begin thide *)
adamc@135	454 (** [[
adamc@135	455 completer'.
adamc@220	456
adamc@205	457 ]]
adamc@205	458
adamc@135	459 Coq loops forever at this point. What went wrong? *)
adamc@220	460
adamc@135	461 Abort.
adamc@141	462 (* end thide *)
adamc@135	463 End firstorder'.
adamc@136	464
adamc@136	465 (** A few examples should illustrate the issue. Here we see a [match]-based proof that works fine: *)
adamc@136	466
adamc@136	467 Theorem t1 : forall x : nat, x = x.
adamc@136	468 match goal with
adamc@136	469 \| [ \|- forall x, _ ] => trivial
adamc@136	470 end.
adamc@141	471 (* begin thide *)
adamc@136	472 Qed.
adamc@141	473 (* end thide *)
adamc@136	474
adamc@136	475 (** This one fails. *)
adamc@136	476
adamc@141	477 (* begin thide *)
adamc@136	478 Theorem t1' : forall x : nat, x = x.
adamc@136	479 (** [[
adamc@136	480 match goal with
adamc@136	481 \| [ \|- forall x, ?P ] => trivial
adamc@136	482 end.
adamc@136	483
adamc@136	484 User error: No matching clauses for match goal
adam@302	485 ]]
adam@302	486 *)
adamc@220	487
adamc@136	488 Abort.
adamc@141	489 (* end thide *)
adamc@136	490
adam@288	491 (** 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. To understand why this applies to the [completer] tactics, recall that, in Coq, implication is shorthand for degenerate universal quantification where the quantified variable is not used. Nonetheless, in an Ltac pattern, Coq is happy to match a wildcard implication against a universal quantification.
adamc@136	492
adam@288	493 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	494
adam@288	495 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, where the same fact may be added to the context repeatedly in an infinite loop. Our earlier [completer] tactic uses a modus ponens rule that matches the implication conclusion with a variable, which blocks matching against non-trivial universal quantifiers. *)
adamc@137	496
adamc@137	497
adamc@137	498 (** * Functional Programming in Ltac *)
adamc@137	499
adamc@141	500 (* EX: Write a list length function in Ltac. *)
adamc@141	501
adamc@137	502 (** 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	503
adamc@137	504 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	505
adamc@137	506 [[
adamc@137	507 Ltac length ls :=
adamc@137	508 match ls with
adamc@137	509 \| nil => O
adamc@137	510 \| _ :: ls' => S (length ls')
adamc@137	511 end.
adamc@137	512
adamc@137	513 Error: The reference ls' was not found in the current environment
adamc@220	514
adamc@137	515 ]]
adamc@137	516
adamc@137	517 At this point, we hopefully remember that pattern variable names must be prefixed by question marks in Ltac.
adamc@137	518
adamc@137	519 [[
adamc@137	520 Ltac length ls :=
adamc@137	521 match ls with
adamc@137	522 \| nil => O
adamc@137	523 \| _ :: ?ls' => S (length ls')
adamc@137	524 end.
adamc@137	525
adamc@137	526 Error: The reference S was not found in the current environment
adamc@220	527
adamc@137	528 ]]
adamc@137	529
adam@288	530 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	531
adamc@141	532 (* begin thide *)
adamc@137	533 Ltac length ls :=
adamc@137	534 match ls with
adamc@137	535 \| nil => O
adamc@137	536 \| _ :: ?ls' => constr:(S (length ls'))
adamc@137	537 end.
adamc@137	538
adamc@137	539 (** This definition is accepted. It can be a little awkward to test Ltac definitions like this. Here is one method. *)
adamc@137	540
adamc@137	541 Goal False.
adamc@137	542 let n := length (1 :: 2 :: 3 :: nil) in
adamc@137	543 pose n.
adamc@137	544 (** [[
adamc@137	545 n := S (length (2 :: 3 :: nil)) : nat
adamc@137	546 ============================
adamc@137	547 False
adamc@220	548
adamc@137	549 ]]
adamc@137	550
adam@301	551 We use the [pose] tactic, which extends the proof context with a new variable that is set equal to a particular 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	552
adamc@220	553 [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	554
adamc@220	555 Abort.
adamc@137	556
adamc@137	557 Reset length.
adamc@137	558
adamc@137	559 (** 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	560
adamc@137	561 Ltac length ls :=
adamc@137	562 match ls with
adamc@137	563 \| nil => O
adamc@137	564 \| _ :: ?ls' =>
adamc@137	565 let ls'' := length ls' in
adamc@137	566 constr:(S ls'')
adamc@137	567 end.
adamc@137	568
adamc@137	569 Goal False.
adamc@137	570 let n := length (1 :: 2 :: 3 :: nil) in
adamc@137	571 pose n.
adamc@137	572 (** [[
adamc@137	573 n := 3 : nat
adamc@137	574 ============================
adamc@137	575 False
adam@302	576 ]]
adam@302	577 *)
adamc@220	578
adamc@137	579 Abort.
adamc@141	580 (* end thide *)
adamc@141	581
adamc@141	582 (* EX: Write a list map function in Ltac. *)
adamc@137	583
adamc@137	584 (** 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	585
adamc@141	586 (* begin thide *)
adamc@137	587 Ltac map T f :=
adamc@137	588 let rec map' ls :=
adamc@137	589 match ls with
adam@288	590 \| nil => constr:( @nil T)
adamc@137	591 \| ?x :: ?ls' =>
adamc@137	592 let x' := f x in
adamc@137	593 let ls'' := map' ls' in
adam@288	594 constr:( x' :: ls'')
adamc@137	595 end in
adamc@137	596 map'.
adamc@137	597
adam@288	598 (** 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	599
adam@288	600 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	601
adamc@137	602 Goal False.
adam@288	603 let ls := map (nat * nat)%type ltac:(fun x => constr:( x, x)) (1 :: 2 :: 3 :: nil) in
adamc@137	604 pose ls.
adamc@137	605 (** [[
adamc@137	606 l := (1, 1) :: (2, 2) :: (3, 3) :: nil : list (nat * nat)
adamc@137	607 ============================
adamc@137	608 False
adam@302	609 ]]
adam@302	610 *)
adamc@220	611
adamc@137	612 Abort.
adamc@141	613 (* end thide *)
adamc@137	614
adamc@138	615
adamc@139	616 (** * Recursive Proof Search *)
adamc@139	617
adamc@139	618 (** 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	619
adam@288	620 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	621
adamc@141	622 (* begin thide *)
adamc@139	623 Ltac inster n :=
adamc@139	624 intuition;
adamc@139	625 match n with
adamc@139	626 \| S ?n' =>
adamc@139	627 match goal with
adamc@139	628 \| [ H : forall x : ?T, _, x : ?T \|- _ ] => generalize (H x); inster n'
adamc@139	629 end
adamc@139	630 end.
adamc@141	631 (* end thide *)
adamc@139	632
adamc@139	633 (** [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	634
adamc@139	635 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	636
adamc@139	637 Section test_inster.
adamc@139	638 Variable A : Set.
adamc@139	639 Variables P Q : A -> Prop.
adamc@139	640 Variable f : A -> A.
adamc@139	641 Variable g : A -> A -> A.
adamc@139	642
adamc@139	643 Hypothesis H1 : forall x y, P (g x y) -> Q (f x).
adamc@139	644
adamc@139	645 Theorem test_inster : forall x y, P (g x y) -> Q (f x).
adamc@220	646 inster 2.
adamc@139	647 Qed.
adamc@139	648
adamc@139	649 Hypothesis H3 : forall u v, P u /\ P v /\ u <> v -> P (g u v).
adamc@139	650 Hypothesis H4 : forall u, Q (f u) -> P u /\ P (f u).
adamc@139	651
adamc@139	652 Theorem test_inster2 : forall x y, x <> y -> P x -> Q (f y) -> Q (f x).
adamc@220	653 inster 3.
adamc@139	654 Qed.
adamc@139	655 End test_inster.
adamc@139	656
adam@288	657 (** 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	658
adam@288	659 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	660
adam@288	661 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	662
adam@288	663 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	664
adamc@138	665 Definition imp (P1 P2 : Prop) := P1 -> P2.
adamc@140	666 Infix "-->" := imp (no associativity, at level 95).
adamc@140	667 Ltac imp := unfold imp; firstorder.
adamc@138	668
adamc@140	669 (** These lemmas about [imp] will be useful in the tactic that we will write. *)
adamc@138	670
adamc@138	671 Theorem and_True_prem : forall P Q,
adamc@138	672 (P /\ True --> Q)
adamc@138	673 -> (P --> Q).
adamc@138	674 imp.
adamc@138	675 Qed.
adamc@138	676
adamc@138	677 Theorem and_True_conc : forall P Q,
adamc@138	678 (P --> Q /\ True)
adamc@138	679 -> (P --> Q).
adamc@138	680 imp.
adamc@138	681 Qed.
adamc@138	682
adamc@138	683 Theorem assoc_prem1 : forall P Q R S,
adamc@138	684 (P /\ (Q /\ R) --> S)
adamc@138	685 -> ((P /\ Q) /\ R --> S).
adamc@138	686 imp.
adamc@138	687 Qed.
adamc@138	688
adamc@138	689 Theorem assoc_prem2 : forall P Q R S,
adamc@138	690 (Q /\ (P /\ R) --> S)
adamc@138	691 -> ((P /\ Q) /\ R --> S).
adamc@138	692 imp.
adamc@138	693 Qed.
adamc@138	694
adamc@138	695 Theorem comm_prem : forall P Q R,
adamc@138	696 (P /\ Q --> R)
adamc@138	697 -> (Q /\ P --> R).
adamc@138	698 imp.
adamc@138	699 Qed.
adamc@138	700
adamc@138	701 Theorem assoc_conc1 : forall P Q R S,
adamc@138	702 (S --> P /\ (Q /\ R))
adamc@138	703 -> (S --> (P /\ Q) /\ R).
adamc@138	704 imp.
adamc@138	705 Qed.
adamc@138	706
adamc@138	707 Theorem assoc_conc2 : forall P Q R S,
adamc@138	708 (S --> Q /\ (P /\ R))
adamc@138	709 -> (S --> (P /\ Q) /\ R).
adamc@138	710 imp.
adamc@138	711 Qed.
adamc@138	712
adamc@138	713 Theorem comm_conc : forall P Q R,
adamc@138	714 (R --> P /\ Q)
adamc@138	715 -> (R --> Q /\ P).
adamc@138	716 imp.
adamc@138	717 Qed.
adamc@138	718
adam@288	719 (** 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	720
adamc@141	721 (* begin thide *)
adamc@138	722 Ltac search_prem tac :=
adamc@138	723 let rec search P :=
adamc@138	724 tac
adamc@138	725 \|\| (apply and_True_prem; tac)
adamc@138	726 \|\| match P with
adamc@138	727 \| ?P1 /\ ?P2 =>
adamc@138	728 (apply assoc_prem1; search P1)
adamc@138	729 \|\| (apply assoc_prem2; search P2)
adamc@138	730 end
adamc@138	731 in match goal with
adamc@138	732 \| [ \|- ?P /\ _ --> _ ] => search P
adamc@138	733 \| [ \|- _ /\ ?P --> _ ] => apply comm_prem; search P
adamc@138	734 \| [ \|- _ --> _ ] => progress (tac \|\| (apply and_True_prem; tac))
adamc@138	735 end.
adamc@138	736
adamc@140	737 (** 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	738
adamc@140	739 [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	740
adamc@140	741 We will also want a dual function [search_conc], which does tree search through an [imp] conclusion. *)
adamc@140	742
adamc@138	743 Ltac search_conc tac :=
adamc@138	744 let rec search P :=
adamc@138	745 tac
adamc@138	746 \|\| (apply and_True_conc; tac)
adamc@138	747 \|\| match P with
adamc@138	748 \| ?P1 /\ ?P2 =>
adamc@138	749 (apply assoc_conc1; search P1)
adamc@138	750 \|\| (apply assoc_conc2; search P2)
adamc@138	751 end
adamc@138	752 in match goal with
adamc@138	753 \| [ \|- _ --> ?P /\ _ ] => search P
adamc@138	754 \| [ \|- _ --> _ /\ ?P ] => apply comm_conc; search P
adamc@138	755 \| [ \|- _ --> _ ] => progress (tac \|\| (apply and_True_conc; tac))
adamc@138	756 end.
adamc@138	757
adamc@140	758 (** 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	759
adamc@138	760 Theorem False_prem : forall P Q,
adamc@138	761 False /\ P --> Q.
adamc@138	762 imp.
adamc@138	763 Qed.
adamc@138	764
adamc@138	765 Theorem True_conc : forall P Q : Prop,
adamc@138	766 (P --> Q)
adamc@138	767 -> (P --> True /\ Q).
adamc@138	768 imp.
adamc@138	769 Qed.
adamc@138	770
adamc@138	771 Theorem Match : forall P Q R : Prop,
adamc@138	772 (Q --> R)
adamc@138	773 -> (P /\ Q --> P /\ R).
adamc@138	774 imp.
adamc@138	775 Qed.
adamc@138	776
adamc@138	777 Theorem ex_prem : forall (T : Type) (P : T -> Prop) (Q R : Prop),
adamc@138	778 (forall x, P x /\ Q --> R)
adamc@138	779 -> (ex P /\ Q --> R).
adamc@138	780 imp.
adamc@138	781 Qed.
adamc@138	782
adamc@138	783 Theorem ex_conc : forall (T : Type) (P : T -> Prop) (Q R : Prop) x,
adamc@138	784 (Q --> P x /\ R)
adamc@138	785 -> (Q --> ex P /\ R).
adamc@138	786 imp.
adamc@138	787 Qed.
adamc@138	788
adam@288	789 (** We will also want a %``%#"#base case#"#%''% lemma for finishing proofs where cancelation has removed every constituent of the conclusion. *)
adamc@140	790
adamc@138	791 Theorem imp_True : forall P,
adamc@138	792 P --> True.
adamc@138	793 imp.
adamc@138	794 Qed.
adamc@138	795
adamc@220	796 (** 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	797
adamc@138	798 Ltac matcher :=
adamc@138	799 intros;
adam@288	800 repeat search_prem ltac:( simple apply False_prem \|\| ( simple apply ex_prem; intro));
adam@288	801 repeat search_conc ltac:( simple apply True_conc \|\| simple eapply ex_conc
adam@288	802 \|\| search_prem ltac:( simple apply Match));
adamc@204	803 try simple apply imp_True.
adamc@141	804 (* end thide *)
adamc@140	805
adamc@140	806 (** Our tactic succeeds at proving a simple example. *)
adamc@138	807
adamc@138	808 Theorem t2 : forall P Q : Prop,
adamc@138	809 Q /\ (P /\ False) /\ P --> P /\ Q.
adamc@138	810 matcher.
adamc@138	811 Qed.
adamc@138	812
adamc@140	813 (** In the generated proof, we find a trace of the workings of the search tactics. *)
adamc@140	814
adamc@140	815 Print t2.
adamc@220	816 (** %\vspace{-.15in}% [[
adamc@140	817 t2 =
adamc@140	818 fun P Q : Prop =>
adamc@140	819 comm_prem (assoc_prem1 (assoc_prem2 (False_prem (P:=P /\ P /\ Q) (P /\ Q))))
adamc@140	820 : forall P Q : Prop, Q /\ (P /\ False) /\ P --> P /\ Q
adamc@220	821
adamc@220	822 ]]
adamc@140	823
adamc@220	824 We can also see that [matcher] is well-suited for cases where some human intervention is needed after the automation finishes. *)
adamc@140	825
adamc@138	826 Theorem t3 : forall P Q R : Prop,
adamc@138	827 P /\ Q --> Q /\ R /\ P.
adamc@138	828 matcher.
adamc@140	829 (** [[
adamc@140	830 ============================
adamc@140	831 True --> R
adamc@220	832
adamc@140	833 ]]
adamc@140	834
adamc@140	835 [matcher] canceled those conjuncts that it was able to cancel, leaving a simplified subgoal for us, much as [intuition] does. *)
adamc@220	836
adamc@138	837 Abort.
adamc@138	838
adamc@140	839 (** [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	840
adamc@138	841 Theorem t4 : forall (P : nat -> Prop) Q, (exists x, P x /\ Q) --> Q /\ (exists x, P x).
adamc@138	842 matcher.
adamc@138	843 Qed.
adamc@138	844
adamc@140	845 Print t4.
adamc@220	846 (** %\vspace{-.15in}% [[
adamc@140	847 t4 =
adamc@140	848 fun (P : nat -> Prop) (Q : Prop) =>
adamc@140	849 and_True_prem
adamc@140	850 (ex_prem (P:=fun x : nat => P x /\ Q)
adamc@140	851 (fun x : nat =>
adamc@140	852 assoc_prem2
adamc@140	853 (Match (P:=Q)
adamc@140	854 (and_True_conc
adamc@140	855 (ex_conc (fun x0 : nat => P x0) x
adamc@140	856 (Match (P:=P x) (imp_True (P:=True))))))))
adamc@140	857 : forall (P : nat -> Prop) (Q : Prop),
adamc@140	858 (exists x : nat, P x /\ Q) --> Q /\ (exists x : nat, P x)
adam@302	859 ]]
adam@302	860 *)
adamc@234	861
adamc@234	862
adamc@234	863 (** * Creating Unification Variables *)
adamc@234	864
adamc@234	865 (** 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	866
adamc@234	867 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	868
adamc@234	869 Before we are ready to write a tactic, we can try out its ingredients one at a time. *)
adamc@234	870
adamc@234	871 Theorem t5 : (forall x : nat, S x > x) -> 2 > 1.
adamc@234	872 intros.
adamc@234	873
adamc@234	874 (** [[
adamc@234	875 H : forall x : nat, S x > x
adamc@234	876 ============================
adamc@234	877 2 > 1
adamc@234	878
adamc@234	879 ]]
adamc@234	880
adamc@234	881 To instantiate [H] generically, we first need to name the value to be used for [x]. *)
adamc@234	882
adamc@234	883 evar (y : nat).
adamc@234	884
adamc@234	885 (** [[
adamc@234	886 H : forall x : nat, S x > x
adamc@234	887 y := ?279 : nat
adamc@234	888 ============================
adamc@234	889 2 > 1
adamc@234	890
adamc@234	891 ]]
adamc@234	892
adamc@234	893 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	894
adamc@234	895 let y' := eval cbv delta [y] in y in
adamc@234	896 clear y; generalize (H y').
adamc@234	897
adamc@234	898 (** [[
adamc@234	899 H : forall x : nat, S x > x
adamc@234	900 ============================
adamc@234	901 S ?279 > ?279 -> 2 > 1
adamc@234	902
adamc@234	903 ]]
adamc@234	904
adamc@234	905 Our instantiation was successful. We can finish by using the refined formula to replace the original. *)
adamc@234	906
adamc@234	907 clear H; intro H.
adamc@234	908
adamc@234	909 (** [[
adamc@234	910 H : S ?281 > ?281
adamc@234	911 ============================
adamc@234	912 2 > 1
adamc@234	913
adamc@234	914 ]]
adamc@234	915
adamc@234	916 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	917
adamc@234	918 apply H.
adamc@234	919 Qed.
adamc@234	920
adamc@234	921 (** Now we can write a tactic that encapsulates the pattern we just employed, instantiating all quantifiers of a particular hypothesis. *)
adamc@234	922
adamc@234	923 Ltac insterU H :=
adamc@234	924 repeat match type of H with
adamc@234	925 \| forall x : ?T, _ =>
adamc@234	926 let x := fresh "x" in
adamc@234	927 evar (x : T);
adamc@234	928 let x' := eval cbv delta [x] in x in
adamc@234	929 clear x; generalize (H x'); clear H; intro H
adamc@234	930 end.
adamc@234	931
adamc@234	932 Theorem t5' : (forall x : nat, S x > x) -> 2 > 1.
adamc@234	933 intro H; insterU H; apply H.
adamc@234	934 Qed.
adamc@234	935
adamc@234	936 (** 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	937
adamc@234	938 Ltac insterKeep H :=
adamc@234	939 let H' := fresh "H'" in
adamc@234	940 generalize H; intro H'; insterU H'.
adamc@234	941
adamc@234	942 Section t6.
adamc@234	943 Variables A B : Type.
adamc@234	944 Variable P : A -> B -> Prop.
adamc@234	945 Variable f : A -> A -> A.
adamc@234	946 Variable g : B -> B -> B.
adamc@234	947
adamc@234	948 Hypothesis H1 : forall v, exists u, P v u.
adamc@234	949 Hypothesis H2 : forall v1 u1 v2 u2,
adamc@234	950 P v1 u1
adamc@234	951 -> P v2 u2
adamc@234	952 -> P (f v1 v2) (g u1 u2).
adamc@234	953
adamc@234	954 Theorem t6 : forall v1 v2, exists u1, exists u2, P (f v1 v2) (g u1 u2).
adamc@234	955 intros.
adamc@234	956
adamc@234	957 (** 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	958
adamc@234	959 do 2 insterKeep H1.
adamc@234	960
adamc@234	961 (** Our proof state is extended with two generic instances of [H1].
adamc@234	962
adamc@234	963 [[
adamc@234	964 H' : exists u : B, P ?4289 u
adamc@234	965 H'0 : exists u : B, P ?4288 u
adamc@234	966 ============================
adamc@234	967 exists u1 : B, exists u2 : B, P (f v1 v2) (g u1 u2)
adamc@234	968
adamc@234	969 ]]
adamc@234	970
adamc@234	971 [eauto] still cannot prove the goal, so we eliminate the two new existential quantifiers. *)
adamc@234	972
adamc@234	973 repeat match goal with
adamc@234	974 \| [ H : ex _ \|- _ ] => destruct H
adamc@234	975 end.
adamc@234	976
adamc@234	977 (** Now the goal is simple enough to solve by logic programming. *)
adamc@234	978
adamc@234	979 eauto.
adamc@234	980 Qed.
adamc@234	981 End t6.
adamc@234	982
adamc@234	983 (** 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	984
adamc@234	985 Section t7.
adamc@234	986 Variables A B : Type.
adamc@234	987 Variable Q : A -> Prop.
adamc@234	988 Variable P : A -> B -> Prop.
adamc@234	989 Variable f : A -> A -> A.
adamc@234	990 Variable g : B -> B -> B.
adamc@234	991
adamc@234	992 Hypothesis H1 : forall v, Q v -> exists u, P v u.
adamc@234	993 Hypothesis H2 : forall v1 u1 v2 u2,
adamc@234	994 P v1 u1
adamc@234	995 -> P v2 u2
adamc@234	996 -> P (f v1 v2) (g u1 u2).
adamc@234	997
adam@297	998 Theorem t7 : forall v1 v2, Q v1 -> Q v2 -> exists u1, exists u2, P (f v1 v2) (g u1 u2).
adamc@234	999 intros; do 2 insterKeep H1;
adamc@234	1000 repeat match goal with
adamc@234	1001 \| [ H : ex _ \|- _ ] => destruct H
adamc@234	1002 end; eauto.
adamc@234	1003
adamc@234	1004 (** This proof script does not hit any errors until the very end, when an error message like this one is displayed.
adamc@234	1005
adamc@234	1006 [[
adamc@234	1007 No more subgoals but non-instantiated existential variables :
adamc@234	1008 Existential 1 =
adamc@234	1009 ?4384 : [A : Type
adamc@234	1010 B : Type
adamc@234	1011 Q : A -> Prop
adamc@234	1012 P : A -> B -> Prop
adamc@234	1013 f : A -> A -> A
adamc@234	1014 g : B -> B -> B
adamc@234	1015 H1 : forall v : A, Q v -> exists u : B, P v u
adamc@234	1016 H2 : forall (v1 : A) (u1 : B) (v2 : A) (u2 : B),
adamc@234	1017 P v1 u1 -> P v2 u2 -> P (f v1 v2) (g u1 u2)
adamc@234	1018 v1 : A
adamc@234	1019 v2 : A
adamc@234	1020 H : Q v1
adamc@234	1021 H0 : Q v2
adamc@234	1022 H' : Q v2 -> exists u : B, P v2 u \|- Q v2]
adamc@234	1023
adamc@234	1024 ]]
adamc@234	1025
adam@288	1026 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	1027
adamc@234	1028 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	1029
adamc@234	1030 Abort.
adamc@234	1031 End t7.
adamc@234	1032
adamc@234	1033 Reset insterU.
adamc@234	1034
adamc@234	1035 (** 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	1036
adamc@234	1037 Ltac insterU tac H :=
adamc@234	1038 repeat match type of H with
adamc@234	1039 \| forall x : ?T, _ =>
adamc@234	1040 match type of T with
adamc@234	1041 \| Prop =>
adamc@234	1042 (let H' := fresh "H'" in
adamc@234	1043 assert (H' : T); [
adamc@234	1044 solve [ tac ]
adamc@234	1045 \| generalize (H H'); clear H H'; intro H ])
adamc@234	1046 \|\| fail 1
adamc@234	1047 \| _ =>
adamc@234	1048 let x := fresh "x" in
adamc@234	1049 evar (x : T);
adamc@234	1050 let x' := eval cbv delta [x] in x in
adamc@234	1051 clear x; generalize (H x'); clear H; intro H
adamc@234	1052 end
adamc@234	1053 end.
adamc@234	1054
adamc@234	1055 Ltac insterKeep tac H :=
adamc@234	1056 let H' := fresh "H'" in
adamc@234	1057 generalize H; intro H'; insterU tac H'.
adamc@234	1058
adamc@234	1059 Section t7.
adamc@234	1060 Variables A B : Type.
adamc@234	1061 Variable Q : A -> Prop.
adamc@234	1062 Variable P : A -> B -> Prop.
adamc@234	1063 Variable f : A -> A -> A.
adamc@234	1064 Variable g : B -> B -> B.
adamc@234	1065
adamc@234	1066 Hypothesis H1 : forall v, Q v -> exists u, P v u.
adamc@234	1067 Hypothesis H2 : forall v1 u1 v2 u2,
adamc@234	1068 P v1 u1
adamc@234	1069 -> P v2 u2
adamc@234	1070 -> P (f v1 v2) (g u1 u2).
adamc@234	1071
adamc@234	1072 Theorem t6 : forall v1 v2, Q v1 -> Q v2 -> exists u1, exists u2, P (f v1 v2) (g u1 u2).
adamc@234	1073
adamc@234	1074 (** 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	1075
adamc@234	1076 intros; do 2 insterKeep ltac:(idtac; match goal with
adamc@234	1077 \| [ H : Q ?v \|- _ ] =>
adamc@234	1078 match goal with
adamc@234	1079 \| [ _ : context[P v _] \|- _ ] => fail 1
adamc@234	1080 \| _ => apply H
adamc@234	1081 end
adamc@234	1082 end) H1;
adamc@234	1083 repeat match goal with
adamc@234	1084 \| [ H : ex _ \|- _ ] => destruct H
adamc@234	1085 end; eauto.
adamc@234	1086 Qed.
adamc@234	1087 End t7.
adamc@234	1088
adamc@234	1089 (** It is often useful to instantiate existential variables explicitly. A built-in tactic provides one way of doing so. *)
adamc@234	1090
adamc@234	1091 Theorem t8 : exists p : nat * nat, fst p = 3.
adamc@234	1092 econstructor; instantiate (1 := (3, 2)); reflexivity.
adamc@234	1093 Qed.
adamc@234	1094
adamc@234	1095 (** 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	1096
adamc@234	1097 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	1098
adamc@234	1099 Ltac equate x y :=
adamc@234	1100 let H := fresh "H" in
adamc@234	1101 assert (H : x = y); [ reflexivity \| clear H ].
adamc@234	1102
adamc@234	1103 (** [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	1104
adamc@234	1105 Theorem t9 : exists p : nat * nat, fst p = 3.
adamc@234	1106 econstructor; match goal with
adamc@234	1107 \| [ \|- fst ?x = 3 ] => equate x (3, 2)
adamc@234	1108 end; reflexivity.
adamc@234	1109 Qed.

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annotate src/Match.v @ 305:690796f4690d