### annotate src/Match.v @ 138:59a2110acf64

Uncommented matcher code
author Adam Chlipala Sun, 26 Oct 2008 16:45:19 -0400 c0bda476b44b a9e90bacbd16
rev   line source
adamc@132 10 (* begin hide *)
adamc@132 16 (* end hide *)
adamc@132 21 \chapter{Proof Search in Ltac}% *)
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 25 (** * Some Built-In Automation Tactics *)
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 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@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@133 34 (** * Hint Databases *)
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 38 The basic hints for [auto] and [eauto] are [Hint Immediate lemma], asking to try solving a goal immediately by applying the premise-free lemma; [Resolve lemma], which does the same but may add new premises that are themselves to be subjects of 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 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 42 Theorem bool_neq : true <> false.
adamc@133 44 (** [crush] would have discharged this goal, but the default hint database for [auto] contains no hint that applies. *)
adamc@133 47 (** 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 49 Hint Extern 1 (_ <> _) => congruence.
adamc@133 51 Theorem bool_neq : true <> false.
adamc@133 55 (** 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 57 [Extern] hints may be implemented with the full Ltac language. This example shows a case where a hint uses a [match]. *)
adamc@133 60 Variable A : Set.
adamc@133 61 Variables P Q : A -> Prop.
adamc@133 63 Hypothesis both : forall x, P x /\ Q x.
adamc@133 65 Theorem forall_and : forall z, P z.
adamc@133 67 (** [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 69 Hint Extern 1 (P ?X) =>
adamc@133 71 | [ H : forall x, P x /\ _ |- _ ] => apply (proj1 (H X))
adamc@133 77 (** 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@133 80 (** After our success on this example, we might get more ambitious and seek to generalize the hint to all possible predicates [P].
adamc@133 83 Hint Extern 1 (?P ?X) =>
adamc@133 85 | [ H : forall x, ?P x /\ _ |- _ ] => apply (proj1 (H X))
adamc@134 92 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@134 94 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 96 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 98 This example shows a direct use of [autorewrite]. *)
adamc@134 101 Variable A : Set.
adamc@134 102 Variable f : A -> A.
adamc@134 104 Hypothesis f_f : forall x, f (f x) = f x.
adamc@134 106 Hint Rewrite f_f : my_db.
adamc@134 108 Lemma f_f_f : forall x, f (f (f x)) = f x.
adamc@134 109 intros; autorewrite with my_db; reflexivity.
adamc@134 112 (** 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@134 115 Variable g : A -> A.
adamc@134 116 Hypothesis f_g : forall x, f x = g x.
adamc@134 117 Hint Rewrite f_g : my_db.
adamc@134 119 Lemma f_f_f' : forall x, f (f (f x)) = f x.
adamc@134 120 intros; autorewrite with my_db.
adamc@134 124 g (g (g x)) = g x
adamc@134 128 (** 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@134 132 (** [autorewrite] works with quantified equalities that include additional premises, but we must be careful to avoid similar incorrect rewritings. *)
adamc@134 135 Variable P : A -> Prop.
adamc@134 136 Variable g : A -> A.
adamc@134 137 Hypothesis f_g : forall x, P x -> f x = g x.
adamc@134 138 Hint Rewrite f_g : my_db.
adamc@134 140 Lemma f_f_f' : forall x, f (f (f x)) = f x.
adamc@134 141 intros; autorewrite with my_db.
adamc@134 145 g (g (g x)) = g x
adamc@134 156 (** The inappropriate rule fired the same three times as before, even though we know we will not be able to prove the premises. *)
adamc@134 160 (** Our final, successful, attempt uses an extra argument to [Hint Rewrite] that specifies a tactic to apply to generated premises. *)
adamc@134 163 Variable P : A -> Prop.
adamc@134 164 Variable g : A -> A.
adamc@134 165 Hypothesis f_g : forall x, P x -> f x = g x.
adamc@134 166 Hint Rewrite f_g using assumption : my_db.
adamc@134 168 Lemma f_f_f' : forall x, f (f (f x)) = f x.
adamc@134 169 intros; autorewrite with my_db; reflexivity.
adamc@134 172 (** [autorewrite] will still use [f_g] when the generated premise is among our assumptions. *)
adamc@134 174 Lemma f_f_f_g : forall x, P x -> f (f x) = g x.
adamc@134 175 intros; autorewrite with my_db; reflexivity.
adamc@134 179 (** 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@134 181 Lemma in_star : forall x y, f (f (f (f x))) = f (f y)
adamc@134 182 -> f x = f (f (f y)).
adamc@134 183 intros; autorewrite with my_db in *; assumption.
adamc@135 189 (** * Ltac Programming Basics *)
adamc@135 191 (** 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@135 193 One common use for [match] tactics is identification of subjects for case analysis, as we see in this tactic definition. *)
adamc@135 197 | [ |- if ?X then _ else _ ] => destruct X
adamc@135 200 (** 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@135 202 Theorem hmm : forall (a b c : bool),
adamc@135 210 intros; repeat find_if; constructor.
adamc@135 213 (** 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@135 215 Another very useful Ltac building block is %\textit{%#<i>#context patterns#</i>#%}%. *)
adamc@135 219 | [ |- context[if ?X then _ else _] ] => destruct X
adamc@135 222 (** 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@135 224 Theorem hmm' : forall (a b c : bool),
adamc@135 232 intros; repeat find_if_inside; constructor.
adamc@135 235 (** We can also use [find_if_inside] to prove goals that [find_if] does not simplify sufficiently. *)
adamc@135 237 Theorem duh2 : forall (a b : bool),
adamc@135 238 (if a then 42 else 42) = (if b then 42 else 42).
adamc@135 239 intros; repeat find_if_inside; reflexivity.
adamc@135 242 (** 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@135 245 repeat match goal with
adamc@135 246 | [ H : ?P |- ?P ] => exact H
adamc@135 248 | [ |- True ] => constructor
adamc@135 249 | [ |- _ /\ _ ] => constructor
adamc@135 250 | [ |- _ -> _ ] => intro
adamc@135 252 | [ H : False |- _ ] => destruct H
adamc@135 253 | [ H : _ /\ _ |- _ ] => destruct H
adamc@135 254 | [ H : _ \/ _ |- _ ] => destruct H
adamc@135 256 | [ H1 : ?P -> ?Q, H2 : ?P |- _ ] =>
adamc@135 257 let H := fresh "H" in
adamc@135 258 generalize (H1 H2); clear H1; intro H
adamc@135 261 (** 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@135 263 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 bind a name for a hypothesis to [destruct].
adamc@135 265 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@135 268 Variables P Q R : Prop.
adamc@138 270 Theorem propositional : (P \/ Q \/ False) /\ (P -> Q) -> True /\ Q.
adamc@135 275 (** 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@135 277 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@135 279 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@135 281 For instance, this (unnecessarily verbose) proof script works: *)
adamc@135 283 Theorem m1 : True.
adamc@135 285 | [ |- _ ] => intro
adamc@135 286 | [ |- True ] => constructor
adamc@135 290 (** 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@135 292 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@135 294 Theorem m2 : forall P Q R : Prop, P -> Q -> R -> Q.
adamc@135 295 intros; match goal with
adamc@135 296 | [ H : _ |- _ ] => pose H
adamc@135 300 r := H1 : R
adamc@135 305 By applying [pose], 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@135 308 | [ H : _ |- _ ] => exact H
adamc@135 312 (** 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@135 314 (** Now we are equipped to implement a tactic for checking that a proposition is not among our hypotheses: *)
adamc@135 316 Ltac notHyp P :=
adamc@135 318 | [ _ : P |- _ ] => fail 1
adamc@135 321 | ?P1 /\ ?P2 => first [ notHyp P1 | notHyp P2 | fail 2 ]
adamc@135 322 | _ => idtac
adamc@135 326 (** 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@135 328 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@135 330 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@135 332 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@135 334 Ltac extend pf :=
adamc@135 335 let t := type of pf in
adamc@135 336 notHyp t; generalize pf; intro.
adamc@135 338 (** 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@135 340 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@135 343 repeat match goal with
adamc@135 344 | [ |- _ /\ _ ] => constructor
adamc@135 345 | [ H : _ /\ _ |- _ ] => destruct H
adamc@135 346 | [ H : ?P -> ?Q, H' : ?P |- _ ] =>
adamc@135 347 generalize (H H'); clear H; intro H
adamc@135 348 | [ |- forall x, _ ] => intro
adamc@135 350 | [ H : forall x, ?P x -> _, H' : ?P ?X |- _ ] =>
adamc@135 351 extend (H X H')
adamc@135 354 (** 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@135 356 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@135 358 We can check that [completer] is working properly: *)
adamc@135 361 Variable A : Set.
adamc@135 362 Variables P Q R S : A -> Prop.
adamc@135 364 Hypothesis H1 : forall x, P x -> Q x /\ R x.
adamc@135 365 Hypothesis H2 : forall x, R x -> S x.
adamc@135 367 Theorem fo : forall x, P x -> S x.
adamc@135 372 H : P x
adamc@135 373 H0 : Q x
adamc@135 374 H3 : R x
adamc@135 375 H4 : S x
adamc@135 384 (** 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@135 387 repeat match goal with
adamc@135 388 | [ |- _ /\ _ ] => constructor
adamc@135 389 | [ H : _ /\ _ |- _ ] => destruct H
adamc@135 390 | [ H : ?P -> _, H' : ?P |- _ ] =>
adamc@135 391 generalize (H H'); clear H; intro H
adamc@135 392 | [ |- forall x, _ ] => intro
adamc@135 394 | [ H : forall x, ?P x -> _, H' : ?P ?X |- _ ] =>
adamc@135 395 extend (H X H')
adamc@135 398 (** 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@135 401 Variable A : Set.
adamc@135 402 Variables P Q R S : A -> Prop.
adamc@135 404 Hypothesis H1 : forall x, P x -> Q x /\ R x.
adamc@135 405 Hypothesis H2 : forall x, R x -> S x.
adamc@135 407 Theorem fo' : forall x, P x -> S x.
adamc@135 412 Coq loops forever at this point. What went wrong? *)
adamc@136 416 (** A few examples should illustrate the issue. Here we see a [match]-based proof that works fine: *)
adamc@136 418 Theorem t1 : forall x : nat, x = x.
adamc@136 420 | [ |- forall x, _ ] => trivial
adamc@136 424 (** This one fails. *)
adamc@136 426 Theorem t1' : forall x : nat, x = x.
adamc@136 430 | [ |- forall x, ?P ] => trivial
adamc@136 434 User error: No matching clauses for match goal
adamc@136 438 (** 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@136 440 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@136 442 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@137 445 (** * Functional Programming in Ltac *)
adamc@137 447 (** 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@137 449 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@137 452 Ltac length ls :=
adamc@137 454 | nil => O
adamc@137 455 | _ :: ls' => S (length ls')
adamc@137 462 At this point, we hopefully remember that pattern variable names must be prefixed by question marks in Ltac.
adamc@137 465 Ltac length ls :=
adamc@137 467 | nil => O
adamc@137 468 | _ :: ?ls' => S (length ls')
adamc@137 475 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@137 477 Ltac length ls :=
adamc@137 479 | nil => O
adamc@137 480 | _ :: ?ls' => constr:(S (length ls'))
adamc@137 483 (** This definition is accepted. It can be a little awkward to test Ltac definitions like this. Here is one method. *)
adamc@137 486 let n := length (1 :: 2 :: 3 :: nil) in
adamc@137 490 n := S (length (2 :: 3 :: nil)) : nat
adamc@137 495 [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. *)Abort.
adamc@137 499 (** 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@137 501 Ltac length ls :=
adamc@137 503 | nil => O
adamc@137 504 | _ :: ?ls' =>
adamc@137 505 let ls'' := length ls' in
adamc@137 510 let n := length (1 :: 2 :: 3 :: nil) in
adamc@137 514 n := 3 : nat
adamc@137 520 (** We can also use anonymous function expressions and local function definitions in Ltac, as this example of a standard list [map] function shows. *)
adamc@137 522 Ltac map T f :=
adamc@137 523 let rec map' ls :=
adamc@137 525 | nil => constr:(@nil T)
adamc@137 526 | ?x :: ?ls' =>
adamc@137 527 let x' := f x in
adamc@137 528 let ls'' := map' ls' in
adamc@137 533 (** 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@137 535 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@137 538 let ls := map (nat * nat)%type ltac:(fun x => constr:(x, x)) (1 :: 2 :: 3 :: nil) in
adamc@137 542 l := (1, 1) :: (2, 2) :: (3, 3) :: nil : list (nat * nat)
adamc@138 549 (** * Proof Search in Continuation-Passing Style *)
adamc@138 551 Definition imp (P1 P2 : Prop) := P1 -> P2.
adamc@138 553 Infix "-->" := imp (no associativity, at level 95).
adamc@138 555 Ltac imp := unfold imp; firstorder.
adamc@138 557 Theorem and_True_prem : forall P Q,
adamc@138 558 (P /\ True --> Q)
adamc@138 559 -> (P --> Q).
adamc@138 563 Theorem and_True_conc : forall P Q,
adamc@138 564 (P --> Q /\ True)
adamc@138 565 -> (P --> Q).
adamc@138 569 Theorem assoc_prem1 : forall P Q R S,
adamc@138 570 (P /\ (Q /\ R) --> S)
adamc@138 571 -> ((P /\ Q) /\ R --> S).
adamc@138 575 Theorem assoc_prem2 : forall P Q R S,
adamc@138 576 (Q /\ (P /\ R) --> S)
adamc@138 577 -> ((P /\ Q) /\ R --> S).
adamc@138 581 Theorem comm_prem : forall P Q R,
adamc@138 582 (P /\ Q --> R)
adamc@138 583 -> (Q /\ P --> R).
adamc@138 587 Theorem assoc_conc1 : forall P Q R S,
adamc@138 588 (S --> P /\ (Q /\ R))
adamc@138 589 -> (S --> (P /\ Q) /\ R).
adamc@138 593 Theorem assoc_conc2 : forall P Q R S,
adamc@138 594 (S --> Q /\ (P /\ R))
adamc@138 595 -> (S --> (P /\ Q) /\ R).
adamc@138 599 Theorem comm_conc : forall P Q R,
adamc@138 600 (R --> P /\ Q)
adamc@138 601 -> (R --> Q /\ P).
adamc@138 605 Ltac search_prem tac :=
adamc@138 606 let rec search P :=
adamc@138 608 || (apply and_True_prem; tac)
adamc@138 609 || match P with
adamc@138 610 | ?P1 /\ ?P2 =>
adamc@138 611 (apply assoc_prem1; search P1)
adamc@138 612 || (apply assoc_prem2; search P2)
adamc@138 614 in match goal with
adamc@138 615 | [ |- ?P /\ _ --> _ ] => search P
adamc@138 616 | [ |- _ /\ ?P --> _ ] => apply comm_prem; search P
adamc@138 617 | [ |- _ --> _ ] => progress (tac || (apply and_True_prem; tac))
adamc@138 620 Ltac search_conc tac :=
adamc@138 621 let rec search P :=
adamc@138 623 || (apply and_True_conc; tac)
adamc@138 624 || match P with
adamc@138 625 | ?P1 /\ ?P2 =>
adamc@138 626 (apply assoc_conc1; search P1)
adamc@138 627 || (apply assoc_conc2; search P2)
adamc@138 629 in match goal with
adamc@138 630 | [ |- _ --> ?P /\ _ ] => search P
adamc@138 631 | [ |- _ --> _ /\ ?P ] => apply comm_conc; search P
adamc@138 632 | [ |- _ --> _ ] => progress (tac || (apply and_True_conc; tac))
adamc@138 635 Theorem False_prem : forall P Q,
adamc@138 636 False /\ P --> Q.
adamc@138 640 Theorem True_conc : forall P Q : Prop,
adamc@138 642 -> (P --> True /\ Q).
adamc@138 646 Theorem Match : forall P Q R : Prop,
adamc@138 648 -> (P /\ Q --> P /\ R).
adamc@138 652 Theorem ex_prem : forall (T : Type) (P : T -> Prop) (Q R : Prop),
adamc@138 653 (forall x, P x /\ Q --> R)
adamc@138 654 -> (ex P /\ Q --> R).
adamc@138 658 Theorem ex_conc : forall (T : Type) (P : T -> Prop) (Q R : Prop) x,
adamc@138 659 (Q --> P x /\ R)
adamc@138 660 -> (Q --> ex P /\ R).
adamc@138 664 Theorem imp_True : forall P,
adamc@138 671 repeat search_prem ltac:(apply False_prem || (apply ex_prem; intro));
adamc@138 672 repeat search_conc ltac:(apply True_conc || eapply ex_conc || search_prem ltac:(apply Match));
adamc@138 675 Theorem t2 : forall P Q : Prop,
adamc@138 676 Q /\ (P /\ False) /\ P --> P /\ Q.