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Build an open-source release of library modules
author Adam Chlipala <adam@chlipala.net>
date Thu, 05 Dec 2013 15:51:33 -0500
parents 49f3b2d70302
children ed829eaa91b2
rev   line source
adam@386 1 (* Copyright (c) 2008-2012, 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
adam@314 13 Require Import CpdtTactics.
adamc@132 14
adamc@132 15 Set Implicit Arguments.
adamc@132 16 (* end hide *)
adamc@132 17
adamc@132 18
adam@324 19 (** %\chapter{Proof Search in Ltac}% *)
adamc@132 20
adam@328 21 (** We have seen many examples of proof automation so far, some with tantalizing code snippets from Ltac, Coq's domain-specific language for proof search procedures. This chapter aims to give a bottom-up presentation of the features of Ltac, focusing in particular on the Ltac %\index{tactics!match}%[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 22
adamc@132 23 (** * Some Built-In Automation Tactics *)
adamc@132 24
adam@386 25 (** A number of tactics are called repeatedly by [crush]. The %\index{tactics!intuition}%[intuition] tactic simplifies propositional structure of goals. The %\index{tactics!congruence}%[congruence] tactic applies the rules of equality and congruence closure, plus properties of constructors of inductive types. The %\index{tactics!omega}%[omega] tactic provides a complete decision procedure for a theory that is called %\index{linear arithmetic}%quantifier-free linear arithmetic or %\index{Presburger arithmetic}%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, with operands built from constants, variables, addition, and subtraction (with multiplication by a constant available as a shorthand for addition or subtraction).
adamc@132 26
adam@411 27 The %\index{tactics!ring}%[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 similar 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 %\index{tactics!fourier}%[fourier] tactic uses Fourier's method to prove inequalities over real numbers, which are axiomatized in the Coq standard library.
adamc@132 28
adam@431 29 The%\index{setoids}% _setoid_ 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."
adam@328 30
adam@431 31 There are several other built-in "black box" automation tactics, which one can learn about by perusing the Coq manual. The real promise of Coq, though, is in the coding of problem-specific tactics with Ltac. *)
adamc@132 32
adamc@132 33
adamc@135 34 (** * Ltac Programming Basics *)
adamc@135 35
adam@328 36 (** We have already seen many examples of Ltac programs. In the rest of this chapter, we attempt to give a thorough introduction to the important features and design patterns.
adamc@135 37
adamc@135 38 One common use for [match] tactics is identification of subjects for case analysis, as we see in this tactic definition. *)
adamc@135 39
adamc@141 40 (* begin thide *)
adamc@135 41 Ltac find_if :=
adamc@135 42 match goal with
adamc@135 43 | [ |- if ?X then _ else _ ] => destruct X
adamc@135 44 end.
adamc@141 45 (* end thide *)
adamc@135 46
adamc@135 47 (** 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 48
adamc@135 49 Theorem hmm : forall (a b c : bool),
adamc@135 50 if a
adamc@135 51 then if b
adamc@135 52 then True
adamc@135 53 else True
adamc@135 54 else if c
adamc@135 55 then True
adamc@135 56 else True.
adamc@141 57 (* begin thide *)
adamc@135 58 intros; repeat find_if; constructor.
adamc@135 59 Qed.
adamc@141 60 (* end thide *)
adamc@135 61
adam@411 62 (** The %\index{tactics!repeat}%[repeat] that we use here is called a%\index{tactical}% _tactical_, or tactic combinator. The behavior of [repeat t] is to loop through running [t], running [t] on all generated subgoals, running [t] on _their_ 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 63
adam@411 64 Another very useful Ltac building block is%\index{context patterns}% _context patterns_. *)
adamc@135 65
adamc@141 66 (* begin thide *)
adamc@135 67 Ltac find_if_inside :=
adamc@135 68 match goal with
adamc@135 69 | [ |- context[if ?X then _ else _] ] => destruct X
adamc@135 70 end.
adamc@141 71 (* end thide *)
adamc@135 72
adamc@135 73 (** 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 74
adamc@135 75 Theorem hmm' : forall (a b c : bool),
adamc@135 76 if a
adamc@135 77 then if b
adamc@135 78 then True
adamc@135 79 else True
adamc@135 80 else if c
adamc@135 81 then True
adamc@135 82 else True.
adamc@141 83 (* begin thide *)
adamc@135 84 intros; repeat find_if_inside; constructor.
adamc@135 85 Qed.
adamc@141 86 (* end thide *)
adamc@135 87
adamc@135 88 (** We can also use [find_if_inside] to prove goals that [find_if] does not simplify sufficiently. *)
adamc@135 89
adamc@141 90 Theorem hmm2 : forall (a b : bool),
adamc@135 91 (if a then 42 else 42) = (if b then 42 else 42).
adamc@141 92 (* begin thide *)
adamc@135 93 intros; repeat find_if_inside; reflexivity.
adamc@135 94 Qed.
adamc@141 95 (* end thide *)
adamc@135 96
adam@431 97 (** 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 98
adamc@141 99 (* begin thide *)
adamc@135 100 Ltac my_tauto :=
adamc@135 101 repeat match goal with
adamc@135 102 | [ H : ?P |- ?P ] => exact H
adamc@135 103
adamc@135 104 | [ |- True ] => constructor
adamc@135 105 | [ |- _ /\ _ ] => constructor
adamc@135 106 | [ |- _ -> _ ] => intro
adamc@135 107
adamc@135 108 | [ H : False |- _ ] => destruct H
adamc@135 109 | [ H : _ /\ _ |- _ ] => destruct H
adamc@135 110 | [ H : _ \/ _ |- _ ] => destruct H
adamc@135 111
adam@328 112 | [ H1 : ?P -> ?Q, H2 : ?P |- _ ] => specialize (H1 H2)
adamc@135 113 end.
adamc@141 114 (* end thide *)
adamc@135 115
adam@328 116 (** 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 %\index{tactics!exact}%[exact] tactic solves a goal completely when given a proof term of the proper type.
adamc@135 117
adam@328 118 It is also trivial to implement the introduction rules (in the sense of %\index{natural deduction}%natural deduction%~\cite{TAPLNatDed}%) 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@135 119
adam@484 120 The last rule implements modus ponens, using a tactic %\index{tactics!specialize}%[specialize] which will replace a hypothesis with a version that is specialized to a provided set of arguments (for quantified variables or local hypotheses from implications). By convention, when the argument to [specialize] is an application of a hypothesis [H] to a set of arguments, the result of the specialization replaces [H]. For other terms, the outcome is the same as with [generalize]. *)
adamc@135 121
adamc@135 122 Section propositional.
adamc@135 123 Variables P Q R : Prop.
adamc@135 124
adamc@138 125 Theorem propositional : (P \/ Q \/ False) /\ (P -> Q) -> True /\ Q.
adamc@141 126 (* begin thide *)
adamc@135 127 my_tauto.
adamc@135 128 Qed.
adamc@141 129 (* end thide *)
adamc@135 130 End propositional.
adamc@135 131
adam@328 132 (** 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@135 133
adamc@135 134 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@135 135
adam@398 136 There is a related pair of two other differences that are much more important than the others. The [match] construct has a _backtracking semantics for failure_. 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@135 137
adamc@135 138 For instance, this (unnecessarily verbose) proof script works: *)
adamc@135 139
adamc@135 140 Theorem m1 : True.
adamc@135 141 match goal with
adamc@135 142 | [ |- _ ] => intro
adamc@135 143 | [ |- True ] => constructor
adamc@135 144 end.
adamc@141 145 (* begin thide *)
adamc@135 146 Qed.
adamc@141 147 (* end thide *)
adamc@135 148
adam@460 149 (** 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, the whole pattern-match would fail. In Coq, we backtrack and try the next pattern, which also matches. Its body tactic succeeds, so the overall tactic succeeds as well.
adamc@135 150
adam@398 151 The example shows how failure can move to a different pattern within a [match]. Failure can also trigger an attempt to find _a different way of matching a single pattern_. Consider another example: *)
adamc@135 152
adamc@135 153 Theorem m2 : forall P Q R : Prop, P -> Q -> R -> Q.
adamc@135 154 intros; match goal with
adamc@220 155 | [ H : _ |- _ ] => idtac H
adamc@135 156 end.
adamc@135 157
adam@431 158 (** Coq prints "[H1]". By applying %\index{tactics!idtac}%[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@135 159
adamc@141 160 (* begin thide *)
adamc@135 161 match goal with
adamc@135 162 | [ H : _ |- _ ] => exact H
adamc@135 163 end.
adamc@135 164 Qed.
adamc@141 165 (* end thide *)
adamc@135 166
adamc@135 167 (** 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@135 168
adamc@135 169 (** Now we are equipped to implement a tactic for checking that a proposition is not among our hypotheses: *)
adamc@135 170
adamc@141 171 (* begin thide *)
adamc@135 172 Ltac notHyp P :=
adamc@135 173 match goal with
adamc@135 174 | [ _ : P |- _ ] => fail 1
adamc@135 175 | _ =>
adamc@135 176 match P with
adamc@135 177 | ?P1 /\ ?P2 => first [ notHyp P1 | notHyp P2 | fail 2 ]
adamc@135 178 | _ => idtac
adamc@135 179 end
adamc@135 180 end.
adamc@141 181 (* end thide *)
adamc@135 182
adam@431 183 (** 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 %\index{tactics!fail}%[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@135 184
adam@328 185 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 %\index{tactics!first}%[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@135 186
adam@328 187 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 %\index{tactics!idtac}%[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@135 188
adamc@135 189 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@135 190
adamc@141 191 (* begin thide *)
adamc@135 192 Ltac extend pf :=
adamc@135 193 let t := type of pf in
adamc@135 194 notHyp t; generalize pf; intro.
adamc@141 195 (* end thide *)
adamc@135 196
adam@386 197 (** We see the useful %\index{tactics!type of}%[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]. The tactic %\index{tactics!generalize}%[generalize] takes as input a term [t] (for instance, a proof of some proposition) and then changes the conclusion from [G] to [T -> G], where [T] is the type of [t] (for instance, the proposition proved by the proof [t]).
adamc@135 198
adam@484 199 With these tactics defined, we can write a tactic [completer] for, among other things, adding to the context all consequences of a set of simple first-order formulas. *)
adamc@135 200
adamc@141 201 (* begin thide *)
adamc@135 202 Ltac completer :=
adamc@135 203 repeat match goal with
adamc@135 204 | [ |- _ /\ _ ] => constructor
adamc@135 205 | [ H : _ /\ _ |- _ ] => destruct H
adam@328 206 | [ H : ?P -> ?Q, H' : ?P |- _ ] => specialize (H H')
adamc@135 207 | [ |- forall x, _ ] => intro
adamc@135 208
adam@328 209 | [ H : forall x, ?P x -> _, H' : ?P ?X |- _ ] => extend (H X H')
adamc@135 210 end.
adamc@141 211 (* end thide *)
adamc@135 212
adamc@135 213 (** 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@135 214
adamc@135 215 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@135 216
adam@483 217 We can check that [completer] is working properly, with a theorem that introduces a spurious variable whose didactic purpose we will come to shortly. *)
adamc@135 218
adamc@135 219 Section firstorder.
adamc@135 220 Variable A : Set.
adamc@135 221 Variables P Q R S : A -> Prop.
adamc@135 222
adamc@135 223 Hypothesis H1 : forall x, P x -> Q x /\ R x.
adamc@135 224 Hypothesis H2 : forall x, R x -> S x.
adamc@135 225
adam@483 226 Theorem fo : forall (y x : A), P x -> S x.
adamc@141 227 (* begin thide *)
adamc@135 228 completer.
adamc@135 229 (** [[
adam@483 230 y : A
adamc@135 231 x : A
adamc@135 232 H : P x
adamc@135 233 H0 : Q x
adamc@135 234 H3 : R x
adamc@135 235 H4 : S x
adamc@135 236 ============================
adamc@135 237 S x
adam@302 238 ]]
adam@302 239 *)
adamc@135 240
adamc@135 241 assumption.
adamc@135 242 Qed.
adamc@141 243 (* end thide *)
adamc@135 244 End firstorder.
adamc@135 245
adam@483 246 (** 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. (We change the second [match] case a bit to make the tactic smart enough to handle some subtleties of Ltac behavior that had not been exercised previously.) *)
adamc@135 247
adamc@141 248 (* begin thide *)
adamc@135 249 Ltac completer' :=
adamc@135 250 repeat match goal with
adamc@135 251 | [ |- _ /\ _ ] => constructor
adam@483 252 | [ H : ?P /\ ?Q |- _ ] => destruct H;
adam@483 253 repeat match goal with
adam@483 254 | [ H' : P /\ Q |- _ ] => clear H'
adam@483 255 end
adam@328 256 | [ H : ?P -> _, H' : ?P |- _ ] => specialize (H H')
adamc@135 257 | [ |- forall x, _ ] => intro
adamc@135 258
adam@328 259 | [ H : forall x, ?P x -> _, H' : ?P ?X |- _ ] => extend (H X H')
adamc@135 260 end.
adamc@141 261 (* end thide *)
adamc@135 262
adam@483 263 (** The only other 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@135 264
adamc@135 265 Section firstorder'.
adamc@135 266 Variable A : Set.
adamc@135 267 Variables P Q R S : A -> Prop.
adamc@135 268
adamc@135 269 Hypothesis H1 : forall x, P x -> Q x /\ R x.
adamc@135 270 Hypothesis H2 : forall x, R x -> S x.
adamc@135 271
adam@483 272 Theorem fo' : forall (y x : A), P x -> S x.
adamc@135 273 completer'.
adam@483 274 (** [[
adam@483 275 y : A
adam@483 276 H1 : P y -> Q y /\ R y
adam@483 277 H2 : R y -> S y
adam@483 278 x : A
adam@483 279 H : P x
adam@483 280 ============================
adam@483 281 S x
adam@483 282 ]]
adam@483 283 The quantified theorems have been instantiated with [y] instead of [x], reducing a provable goal to one that is unprovable. Our code in the last [match] case for [completer'] is careful only to instantiate quantifiers along with suitable hypotheses, so why were incorrect choices made?
adam@483 284 *)
adamc@220 285
adamc@135 286 Abort.
adamc@141 287 (* end thide *)
adamc@135 288 End firstorder'.
adamc@136 289
adamc@136 290 (** A few examples should illustrate the issue. Here we see a [match]-based proof that works fine: *)
adamc@136 291
adamc@136 292 Theorem t1 : forall x : nat, x = x.
adamc@136 293 match goal with
adamc@136 294 | [ |- forall x, _ ] => trivial
adamc@136 295 end.
adamc@141 296 (* begin thide *)
adamc@136 297 Qed.
adamc@141 298 (* end thide *)
adamc@136 299
adamc@136 300 (** This one fails. *)
adamc@136 301
adamc@141 302 (* begin thide *)
adamc@136 303 Theorem t1' : forall x : nat, x = x.
adam@445 304 (** %\vspace{-.25in}%[[
adamc@136 305 match goal with
adamc@136 306 | [ |- forall x, ?P ] => trivial
adamc@136 307 end.
adam@328 308 ]]
adamc@136 309
adam@328 310 <<
adamc@136 311 User error: No matching clauses for match goal
adam@328 312 >>
adam@328 313 *)
adamc@220 314
adamc@136 315 Abort.
adamc@141 316 (* end thide *)
adamc@136 317
adam@507 318 (** 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 restriction affects the behavior of the [completer] tactic, 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@136 319
adam@484 320 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. We will see an example of this fancier binding form in Section 15.5.
adamc@136 321
adam@483 322 No matter which Coq version you use, it is important to be aware of this restriction. As we have alluded to, the restriction is the culprit behind the surprising behavior of [completer']. We unintentionally match quantified facts with the modus ponens rule, circumventing the check that a suitably matching hypothesis is available and leading to different behavior, where wrong quantifier instantiations are chosen. 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.
adam@483 323
adam@483 324 Actually, the behavior demonstrated here applies to Coq version 8.4, but not 8.4pl1. The latter version will allow regular Ltac pattern variables to match terms that contain locally bound variables, but a tactic failure occurs if that variable is later used as a Gallina term. *)
adamc@137 325
adamc@137 326
adamc@137 327 (** * Functional Programming in Ltac *)
adamc@137 328
adamc@141 329 (* EX: Write a list length function in Ltac. *)
adamc@141 330
adamc@137 331 (** 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@137 332
adam@475 333 To illustrate, let us try to write a simple list length function. We start out writing it just as in Gallina, simply replacing [Fixpoint] (and its annotations) with [Ltac].
adamc@137 334 [[
adamc@137 335 Ltac length ls :=
adamc@137 336 match ls with
adamc@137 337 | nil => O
adamc@137 338 | _ :: ls' => S (length ls')
adamc@137 339 end.
adam@328 340 ]]
adamc@137 341
adam@328 342 <<
adamc@137 343 Error: The reference ls' was not found in the current environment
adam@328 344 >>
adamc@137 345
adamc@137 346 At this point, we hopefully remember that pattern variable names must be prefixed by question marks in Ltac.
adamc@137 347 [[
adamc@137 348 Ltac length ls :=
adamc@137 349 match ls with
adamc@137 350 | nil => O
adamc@137 351 | _ :: ?ls' => S (length ls')
adamc@137 352 end.
adamc@137 353 ]]
adamc@137 354
adam@328 355 <<
adam@328 356 Error: The reference S was not found in the current environment
adam@328 357 >>
adam@328 358
adam@431 359 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.%\index{tactics!constr}% *)
adamc@137 360
adamc@141 361 (* begin thide *)
adamc@137 362 Ltac length ls :=
adamc@137 363 match ls with
adamc@137 364 | nil => O
adamc@137 365 | _ :: ?ls' => constr:(S (length ls'))
adamc@137 366 end.
adamc@137 367
adam@445 368 (** This definition is accepted. It can be a little awkward to test Ltac definitions like this one. Here is one method. *)
adamc@137 369
adamc@137 370 Goal False.
adamc@137 371 let n := length (1 :: 2 :: 3 :: nil) in
adamc@137 372 pose n.
adamc@137 373 (** [[
adamc@137 374 n := S (length (2 :: 3 :: nil)) : nat
adamc@137 375 ============================
adamc@137 376 False
adamc@137 377 ]]
adamc@137 378
adam@328 379 We use the %\index{tactics!pose}%[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@220 380
adam@328 381 The value of [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@220 382
adamc@220 383 Abort.
adamc@137 384
adamc@137 385 Reset length.
adamc@137 386
adamc@137 387 (** 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@137 388
adamc@137 389 Ltac length ls :=
adamc@137 390 match ls with
adamc@137 391 | nil => O
adamc@137 392 | _ :: ?ls' =>
adamc@137 393 let ls'' := length ls' in
adamc@137 394 constr:(S ls'')
adamc@137 395 end.
adamc@137 396
adamc@137 397 Goal False.
adamc@137 398 let n := length (1 :: 2 :: 3 :: nil) in
adamc@137 399 pose n.
adamc@137 400 (** [[
adamc@137 401 n := 3 : nat
adamc@137 402 ============================
adamc@137 403 False
adam@302 404 ]]
adam@302 405 *)
adamc@220 406
adamc@137 407 Abort.
adamc@141 408 (* end thide *)
adamc@141 409
adamc@141 410 (* EX: Write a list map function in Ltac. *)
adamc@137 411
adam@431 412 (* begin hide *)
adam@437 413 (* begin thide *)
adam@431 414 Definition mapp := (map, list).
adam@437 415 (* end thide *)
adam@431 416 (* end hide *)
adam@431 417
adamc@137 418 (** We can also use anonymous function expressions and local function definitions in Ltac, as this example of a standard list [map] function shows. *)
adamc@137 419
adamc@141 420 (* begin thide *)
adamc@137 421 Ltac map T f :=
adamc@137 422 let rec map' ls :=
adamc@137 423 match ls with
adam@411 424 | nil => constr:(@nil T)
adamc@137 425 | ?x :: ?ls' =>
adamc@137 426 let x' := f x in
adamc@137 427 let ls'' := map' ls' in
adam@411 428 constr:(x' :: ls'')
adamc@137 429 end in
adamc@137 430 map'.
adamc@137 431
adam@411 432 (** 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. The function [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@137 433
adam@431 434 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 %\coqdocvar{%#<tt>#map#</tt>#%}%. *)
adamc@137 435
adamc@137 436 Goal False.
adam@411 437 let ls := map (nat * nat)%type ltac:(fun x => constr:(x, x)) (1 :: 2 :: 3 :: nil) in
adamc@137 438 pose ls.
adamc@137 439 (** [[
adamc@137 440 l := (1, 1) :: (2, 2) :: (3, 3) :: nil : list (nat * nat)
adamc@137 441 ============================
adamc@137 442 False
adam@302 443 ]]
adam@302 444 *)
adamc@220 445
adamc@137 446 Abort.
adamc@141 447 (* end thide *)
adamc@137 448
adam@431 449 (** Each position within an Ltac script has a default applicable non-terminal, where [constr] and [ltac] are the main options worth thinking about, standing respectively for terms of Gallina and Ltac. The explicit colon notation can always be used to override the default non-terminal choice, though code being parsed as Gallina can no longer use such overrides. Within the [ltac] non-terminal, top-level function applications are treated as applications in Ltac, not Gallina; but the _arguments_ to such functions are parsed with [constr] by default. This choice may seem strange, until we realize that we have been relying on it all along in all the proof scripts we write! For instance, the [apply] tactic is an Ltac function, and it is natural to interpret its argument as a term of Gallina, not Ltac. We use an [ltac] prefix to parse Ltac function arguments as Ltac terms themselves, as in the call to %\coqdocvar{%#<tt>#map#</tt>#%}% above. For some simple cases, Ltac terms may be passed without an extra prefix. For instance, an identifier that has an Ltac meaning but no Gallina meaning will be interpreted in Ltac automatically.
adam@386 450
adam@431 451 One other gotcha shows up when we want to debug our Ltac functional programs. We might expect the following code to work, to give us a version of %\coqdocvar{%#<tt>#length#</tt>#%}% that prints a debug trace of the arguments it is called with. *)
adam@328 452
adam@334 453 (* begin thide *)
adam@328 454 Reset length.
adam@328 455
adam@328 456 Ltac length ls :=
adam@328 457 idtac ls;
adam@328 458 match ls with
adam@328 459 | nil => O
adam@328 460 | _ :: ?ls' =>
adam@328 461 let ls'' := length ls' in
adam@328 462 constr:(S ls'')
adam@328 463 end.
adam@328 464
adam@328 465 (** Coq accepts the tactic definition, but the code is fatally flawed and will always lead to dynamic type errors. *)
adam@328 466
adam@328 467 Goal False.
adam@328 468 (** %\vspace{-.15in}%[[
adam@328 469 let n := length (1 :: 2 :: 3 :: nil) in
adam@328 470 pose n.
adam@328 471 ]]
adam@328 472
adam@328 473 <<
adam@328 474 Error: variable n should be bound to a term.
adam@328 475 >> *)
adam@328 476 Abort.
adam@328 477
adam@475 478 (** What is going wrong here? The answer has to do with the dual status of Ltac as both a purely functional and an imperative programming language. The basic programming language is purely functional, but tactic scripts are one "datatype" that can be returned by such programs, and Coq will run such a script using an imperative semantics that mutates proof states. Readers familiar with %\index{monad}\index{Haskell}%monadic programming in Haskell%~\cite{Monads,IO}% may recognize a similarity. Haskell programs with side effects can be thought of as pure programs that return _the code of programs in an imperative language_, where some out-of-band mechanism takes responsibility for running these derived programs. In this way, Haskell remains pure, while supporting usual input-output side effects and more. Ltac uses the same basic mechanism, but in a dynamically typed setting. Here the embedded imperative language includes all the tactics we have been applying so far.
adam@328 479
adam@328 480 Even basic [idtac] is an embedded imperative program, so we may not automatically mix it with purely functional code. In fact, a semicolon operator alone marks a span of Ltac code as an embedded tactic script. This makes some amount of sense, since pure functional languages have no need for sequencing: since they lack side effects, there is no reason to run an expression and then just throw away its value and move on to another expression.
adam@328 481
adam@484 482 An alternate explanation that avoids an analogy to Haskell monads (admittedly a tricky concept in its own right) is: An Ltac tactic program returns a function that, when run later, will perform the desired proof modification. These functions are distinct from other types of data, like numbers or Gallina terms. The prior, correctly working version of [length] computed solely with Gallina terms, but the new one is implicitly returning a tactic function, as indicated by the use of [idtac] and semicolon. However, the new version's recursive call to [length] is structured to expect a Gallina term, not a tactic function, as output. As a result, we have a basic dynamic type error, perhaps obscured by the involvement of first-class tactic scripts.
adam@484 483
adam@431 484 The solution is like in Haskell: we must "monadify" our pure program to give it access to side effects. The trouble is that the embedded tactic language has no [return] construct. Proof scripts are about proving theorems, not calculating results. We can apply a somewhat awkward workaround that requires translating our program into%\index{continuation-passing style}% _continuation-passing style_ %\cite{continuations}%, a program structuring idea popular in functional programming. *)
adam@328 485
adam@328 486 Reset length.
adam@328 487
adam@328 488 Ltac length ls k :=
adam@328 489 idtac ls;
adam@328 490 match ls with
adam@328 491 | nil => k O
adam@328 492 | _ :: ?ls' => length ls' ltac:(fun n => k (S n))
adam@328 493 end.
adam@334 494 (* end thide *)
adam@328 495
adam@431 496 (** The new [length] takes a new input: a _continuation_ [k], which is a function to be called to continue whatever proving process we were in the middle of when we called %\coqdocvar{%#<tt>#length#</tt>#%}%. The argument passed to [k] may be thought of as the return value of %\coqdocvar{%#<tt>#length#</tt>#%}%. *)
adam@328 497
adam@334 498 (* begin thide *)
adam@328 499 Goal False.
adam@328 500 length (1 :: 2 :: 3 :: nil) ltac:(fun n => pose n).
adam@328 501 (** [[
adam@328 502 (1 :: 2 :: 3 :: nil)
adam@328 503 (2 :: 3 :: nil)
adam@328 504 (3 :: nil)
adam@328 505 nil
adam@328 506 ]]
adam@328 507 *)
adam@328 508 Abort.
adam@334 509 (* end thide *)
adam@328 510
adam@386 511 (** We see exactly the trace of function arguments that we expected initially, and an examination of the proof state afterward would show that variable [n] has been added with value [3].
adam@386 512
adam@431 513 Considering the comparison with Haskell's IO monad, there is an important subtlety that deserves to be mentioned. A Haskell IO computation represents (theoretically speaking, at least) a transformer from one state of the real world to another, plus a pure value to return. Some of the state can be very specific to the program, as in the case of heap-allocated mutable references, but some can be along the lines of the favorite example "launch missile," where the program has a side effect on the real world that is not possible to undo.
adam@386 514
adam@398 515 In contrast, Ltac scripts can be thought of as controlling just two simple kinds of mutable state. First, there is the current sequence of proof subgoals. Second, there is a partial assignment of discovered values to unification variables introduced by proof search (for instance, by [eauto], as we saw in the previous chapter). Crucially, _every mutation of this state can be undone_ during backtracking introduced by [match], [auto], and other built-in Ltac constructs. Ltac proof scripts have state, but it is purely local, and all changes to it are reversible, which is a very useful semantics for proof search. *)
adam@328 516
adamc@138 517
adamc@139 518 (** * Recursive Proof Search *)
adamc@139 519
adamc@139 520 (** 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@139 521
adam@431 522 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@139 523
adamc@141 524 (* begin thide *)
adamc@139 525 Ltac inster n :=
adamc@139 526 intuition;
adamc@139 527 match n with
adamc@139 528 | S ?n' =>
adamc@139 529 match goal with
adam@460 530 | [ H : forall x : ?T, _, y : ?T |- _ ] => generalize (H y); inster n'
adamc@139 531 end
adamc@139 532 end.
adamc@141 533 (* end thide *)
adamc@139 534
adam@507 535 (** The tactic begins by applying propositional simplification. Next, it checks if any chain length remains, failing if not. Otherwise, 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@139 536
adamc@139 537 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@139 538
adamc@139 539 Section test_inster.
adamc@139 540 Variable A : Set.
adamc@139 541 Variables P Q : A -> Prop.
adamc@139 542 Variable f : A -> A.
adamc@139 543 Variable g : A -> A -> A.
adamc@139 544
adamc@139 545 Hypothesis H1 : forall x y, P (g x y) -> Q (f x).
adamc@139 546
adam@328 547 Theorem test_inster : forall x, P (g x x) -> Q (f x).
adamc@220 548 inster 2.
adamc@139 549 Qed.
adamc@139 550
adamc@139 551 Hypothesis H3 : forall u v, P u /\ P v /\ u <> v -> P (g u v).
adamc@139 552 Hypothesis H4 : forall u, Q (f u) -> P u /\ P (f u).
adamc@139 553
adamc@139 554 Theorem test_inster2 : forall x y, x <> y -> P x -> Q (f y) -> Q (f x).
adamc@220 555 inster 3.
adamc@139 556 Qed.
adamc@139 557 End test_inster.
adamc@139 558
adam@431 559 (** 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, recalling the discussion at the end of the last section, 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@140 560
adam@431 561 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@140 562
adam@431 563 This procedure is inspired by one for separation logic%~\cite{separation}%, 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@140 564
adam@431 565 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@138 566
adamc@138 567 Definition imp (P1 P2 : Prop) := P1 -> P2.
adamc@140 568 Infix "-->" := imp (no associativity, at level 95).
adamc@140 569 Ltac imp := unfold imp; firstorder.
adamc@138 570
adamc@140 571 (** These lemmas about [imp] will be useful in the tactic that we will write. *)
adamc@138 572
adamc@138 573 Theorem and_True_prem : forall P Q,
adamc@138 574 (P /\ True --> Q)
adamc@138 575 -> (P --> Q).
adamc@138 576 imp.
adamc@138 577 Qed.
adamc@138 578
adamc@138 579 Theorem and_True_conc : forall P Q,
adamc@138 580 (P --> Q /\ True)
adamc@138 581 -> (P --> Q).
adamc@138 582 imp.
adamc@138 583 Qed.
adamc@138 584
adam@488 585 Theorem pick_prem1 : forall P Q R S,
adamc@138 586 (P /\ (Q /\ R) --> S)
adamc@138 587 -> ((P /\ Q) /\ R --> S).
adamc@138 588 imp.
adamc@138 589 Qed.
adamc@138 590
adam@488 591 Theorem pick_prem2 : forall P Q R S,
adamc@138 592 (Q /\ (P /\ R) --> S)
adamc@138 593 -> ((P /\ Q) /\ R --> S).
adamc@138 594 imp.
adamc@138 595 Qed.
adamc@138 596
adamc@138 597 Theorem comm_prem : forall P Q R,
adamc@138 598 (P /\ Q --> R)
adamc@138 599 -> (Q /\ P --> R).
adamc@138 600 imp.
adamc@138 601 Qed.
adamc@138 602
adam@488 603 Theorem pick_conc1 : forall P Q R S,
adamc@138 604 (S --> P /\ (Q /\ R))
adamc@138 605 -> (S --> (P /\ Q) /\ R).
adamc@138 606 imp.
adamc@138 607 Qed.
adamc@138 608
adam@488 609 Theorem pick_conc2 : forall P Q R S,
adamc@138 610 (S --> Q /\ (P /\ R))
adamc@138 611 -> (S --> (P /\ Q) /\ R).
adamc@138 612 imp.
adamc@138 613 Qed.
adamc@138 614
adamc@138 615 Theorem comm_conc : forall P Q R,
adamc@138 616 (R --> P /\ Q)
adamc@138 617 -> (R --> Q /\ P).
adamc@138 618 imp.
adamc@138 619 Qed.
adamc@138 620
adam@431 621 (** 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@140 622
adamc@138 623 Ltac search_prem tac :=
adamc@138 624 let rec search P :=
adamc@138 625 tac
adamc@138 626 || (apply and_True_prem; tac)
adamc@138 627 || match P with
adamc@138 628 | ?P1 /\ ?P2 =>
adam@488 629 (apply pick_prem1; search P1)
adam@488 630 || (apply pick_prem2; search P2)
adamc@138 631 end
adamc@138 632 in match goal with
adamc@138 633 | [ |- ?P /\ _ --> _ ] => search P
adamc@138 634 | [ |- _ /\ ?P --> _ ] => apply comm_prem; search P
adamc@138 635 | [ |- _ --> _ ] => progress (tac || (apply and_True_prem; tac))
adamc@138 636 end.
adamc@138 637
adam@460 638 (** 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. The call [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. The %\index{tactics!progress}%[progress] tactical fails when its argument tactic succeeds without changing the current subgoal.
adamc@140 639
adam@507 640 The [search] function itself tries the same tricks as in the last case of the final [match], using the [||] operator as a shorthand for trying one tactic and then, if the first fails, trying another. Additionally, if neither works, it checks if [P] is a conjunction. If so, it calls itself recursively on each conjunct, first applying associativity/commutativity lemmas to maintain the goal-form invariant.
adamc@140 641
adamc@140 642 We will also want a dual function [search_conc], which does tree search through an [imp] conclusion. *)
adamc@140 643
adamc@138 644 Ltac search_conc tac :=
adamc@138 645 let rec search P :=
adamc@138 646 tac
adamc@138 647 || (apply and_True_conc; tac)
adamc@138 648 || match P with
adamc@138 649 | ?P1 /\ ?P2 =>
adam@488 650 (apply pick_conc1; search P1)
adam@488 651 || (apply pick_conc2; search P2)
adamc@138 652 end
adamc@138 653 in match goal with
adamc@138 654 | [ |- _ --> ?P /\ _ ] => search P
adamc@138 655 | [ |- _ --> _ /\ ?P ] => apply comm_conc; search P
adamc@138 656 | [ |- _ --> _ ] => progress (tac || (apply and_True_conc; tac))
adamc@138 657 end.
adamc@138 658
adamc@140 659 (** 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@140 660
adam@328 661 (* begin thide *)
adamc@138 662 Theorem False_prem : forall P Q,
adamc@138 663 False /\ P --> Q.
adamc@138 664 imp.
adamc@138 665 Qed.
adamc@138 666
adamc@138 667 Theorem True_conc : forall P Q : Prop,
adamc@138 668 (P --> Q)
adamc@138 669 -> (P --> True /\ Q).
adamc@138 670 imp.
adamc@138 671 Qed.
adamc@138 672
adamc@138 673 Theorem Match : forall P Q R : Prop,
adamc@138 674 (Q --> R)
adamc@138 675 -> (P /\ Q --> P /\ R).
adamc@138 676 imp.
adamc@138 677 Qed.
adamc@138 678
adamc@138 679 Theorem ex_prem : forall (T : Type) (P : T -> Prop) (Q R : Prop),
adamc@138 680 (forall x, P x /\ Q --> R)
adamc@138 681 -> (ex P /\ Q --> R).
adamc@138 682 imp.
adamc@138 683 Qed.
adamc@138 684
adamc@138 685 Theorem ex_conc : forall (T : Type) (P : T -> Prop) (Q R : Prop) x,
adamc@138 686 (Q --> P x /\ R)
adamc@138 687 -> (Q --> ex P /\ R).
adamc@138 688 imp.
adamc@138 689 Qed.
adamc@138 690
adam@465 691 (** We will also want a "base case" lemma for finishing proofs where cancellation has removed every constituent of the conclusion. *)
adamc@140 692
adamc@138 693 Theorem imp_True : forall P,
adamc@138 694 P --> True.
adamc@138 695 imp.
adamc@138 696 Qed.
adamc@138 697
adam@386 698 (** 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 %\index{tactics!simple apply}%[simple apply] in place of [apply] to use a simpler, less expensive unification algorithm. *)
adamc@140 699
adamc@138 700 Ltac matcher :=
adamc@138 701 intros;
adam@411 702 repeat search_prem ltac:(simple apply False_prem || (simple apply ex_prem; intro));
adam@411 703 repeat search_conc ltac:(simple apply True_conc || simple eapply ex_conc
adam@411 704 || search_prem ltac:(simple apply Match));
adamc@204 705 try simple apply imp_True.
adamc@141 706 (* end thide *)
adamc@140 707
adamc@140 708 (** Our tactic succeeds at proving a simple example. *)
adamc@138 709
adamc@138 710 Theorem t2 : forall P Q : Prop,
adamc@138 711 Q /\ (P /\ False) /\ P --> P /\ Q.
adamc@138 712 matcher.
adamc@138 713 Qed.
adamc@138 714
adamc@140 715 (** In the generated proof, we find a trace of the workings of the search tactics. *)
adamc@140 716
adamc@140 717 Print t2.
adamc@220 718 (** %\vspace{-.15in}% [[
adamc@140 719 t2 =
adamc@140 720 fun P Q : Prop =>
adam@488 721 comm_prem (pick_prem1 (pick_prem2 (False_prem (P:=P /\ P /\ Q) (P /\ Q))))
adamc@140 722 : forall P Q : Prop, Q /\ (P /\ False) /\ P --> P /\ Q
adamc@220 723 ]]
adamc@140 724
adam@460 725 %\smallskip{}%We can also see that [matcher] is well-suited for cases where some human intervention is needed after the automation finishes. *)
adamc@140 726
adamc@138 727 Theorem t3 : forall P Q R : Prop,
adamc@138 728 P /\ Q --> Q /\ R /\ P.
adamc@138 729 matcher.
adamc@140 730 (** [[
adamc@140 731 ============================
adamc@140 732 True --> R
adamc@220 733
adamc@140 734 ]]
adamc@140 735
adam@328 736 Our tactic canceled those conjuncts that it was able to cancel, leaving a simplified subgoal for us, much as [intuition] does. *)
adamc@220 737
adamc@138 738 Abort.
adamc@138 739
adam@328 740 (** The [matcher] tactic 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@140 741
adamc@138 742 Theorem t4 : forall (P : nat -> Prop) Q, (exists x, P x /\ Q) --> Q /\ (exists x, P x).
adamc@138 743 matcher.
adamc@138 744 Qed.
adamc@138 745
adamc@140 746 Print t4.
adamc@220 747 (** %\vspace{-.15in}% [[
adamc@140 748 t4 =
adamc@140 749 fun (P : nat -> Prop) (Q : Prop) =>
adamc@140 750 and_True_prem
adamc@140 751 (ex_prem (P:=fun x : nat => P x /\ Q)
adamc@140 752 (fun x : nat =>
adam@488 753 pick_prem2
adamc@140 754 (Match (P:=Q)
adamc@140 755 (and_True_conc
adamc@140 756 (ex_conc (fun x0 : nat => P x0) x
adamc@140 757 (Match (P:=P x) (imp_True (P:=True))))))))
adamc@140 758 : forall (P : nat -> Prop) (Q : Prop),
adamc@140 759 (exists x : nat, P x /\ Q) --> Q /\ (exists x : nat, P x)
adam@302 760 ]]
adam@386 761
adam@386 762 This proof term is a mouthful, and we can be glad that we did not build it manually! *)
adamc@234 763
adamc@234 764
adamc@234 765 (** * Creating Unification Variables *)
adamc@234 766
adam@398 767 (** A final useful ingredient in tactic crafting is the ability to allocate new unification variables explicitly. Tactics like [eauto] introduce unification variables internally to support flexible proof search. While [eauto] and its relatives do _backward_ reasoning, we often want to do similar _forward_ reasoning, where unification variables can be useful for similar reasons.
adamc@234 768
adam@465 769 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 instantiations 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@234 770
adamc@234 771 Before we are ready to write a tactic, we can try out its ingredients one at a time. *)
adamc@234 772
adamc@234 773 Theorem t5 : (forall x : nat, S x > x) -> 2 > 1.
adamc@234 774 intros.
adamc@234 775
adamc@234 776 (** [[
adamc@234 777 H : forall x : nat, S x > x
adamc@234 778 ============================
adamc@234 779 2 > 1
adamc@234 780
adamc@234 781 ]]
adamc@234 782
adam@328 783 To instantiate [H] generically, we first need to name the value to be used for [x].%\index{tactics!evar}% *)
adamc@234 784
adamc@234 785 evar (y : nat).
adamc@234 786
adamc@234 787 (** [[
adamc@234 788 H : forall x : nat, S x > x
adamc@234 789 y := ?279 : nat
adamc@234 790 ============================
adamc@234 791 2 > 1
adamc@234 792
adamc@234 793 ]]
adamc@234 794
adam@328 795 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 unfolding in the expression [y], using the %\index{tactics!eval}%[eval] Ltac construct, which works with the same reduction strategies that we have seen in tactics (e.g., [simpl], [compute], etc.). *)
adamc@234 796
adam@328 797 let y' := eval unfold y in y in
adam@386 798 clear y; specialize (H y').
adamc@234 799
adamc@234 800 (** [[
adam@386 801 H : S ?279 > ?279
adamc@234 802 ============================
adam@386 803 2 > 1
adamc@234 804
adamc@234 805 ]]
adamc@234 806
adam@386 807 Our instantiation was successful. We can finish the proof by using [apply]'s unification to figure out the proper value of [?279]. *)
adamc@234 808
adamc@234 809 apply H.
adamc@234 810 Qed.
adamc@234 811
adamc@234 812 (** Now we can write a tactic that encapsulates the pattern we just employed, instantiating all quantifiers of a particular hypothesis. *)
adamc@234 813
adamc@234 814 Ltac insterU H :=
adamc@234 815 repeat match type of H with
adamc@234 816 | forall x : ?T, _ =>
adamc@234 817 let x := fresh "x" in
adamc@234 818 evar (x : T);
adam@328 819 let x' := eval unfold x in x in
adam@328 820 clear x; specialize (H x')
adamc@234 821 end.
adamc@234 822
adamc@234 823 Theorem t5' : (forall x : nat, S x > x) -> 2 > 1.
adamc@234 824 intro H; insterU H; apply H.
adamc@234 825 Qed.
adamc@234 826
adam@328 827 (** 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. We use the Ltac construct %\index{tactics!fresh}%[fresh] to generate a hypothesis name that is not already used, based on a string suggesting a good name. *)
adamc@234 828
adamc@234 829 Ltac insterKeep H :=
adamc@234 830 let H' := fresh "H'" in
adamc@234 831 generalize H; intro H'; insterU H'.
adamc@234 832
adamc@234 833 Section t6.
adamc@234 834 Variables A B : Type.
adamc@234 835 Variable P : A -> B -> Prop.
adamc@234 836 Variable f : A -> A -> A.
adamc@234 837 Variable g : B -> B -> B.
adamc@234 838
adamc@234 839 Hypothesis H1 : forall v, exists u, P v u.
adamc@234 840 Hypothesis H2 : forall v1 u1 v2 u2,
adamc@234 841 P v1 u1
adamc@234 842 -> P v2 u2
adamc@234 843 -> P (f v1 v2) (g u1 u2).
adamc@234 844
adamc@234 845 Theorem t6 : forall v1 v2, exists u1, exists u2, P (f v1 v2) (g u1 u2).
adamc@234 846 intros.
adamc@234 847
adam@328 848 (** Neither [eauto] nor [firstorder] is clever enough to prove this goal. We can help out by doing some of the work with quantifiers ourselves, abbreviating the proof with the %\index{tactics!do}%[do] tactical for repetition of a tactic a set number of times. *)
adamc@234 849
adamc@234 850 do 2 insterKeep H1.
adamc@234 851
adamc@234 852 (** Our proof state is extended with two generic instances of [H1].
adamc@234 853
adamc@234 854 [[
adamc@234 855 H' : exists u : B, P ?4289 u
adamc@234 856 H'0 : exists u : B, P ?4288 u
adamc@234 857 ============================
adamc@234 858 exists u1 : B, exists u2 : B, P (f v1 v2) (g u1 u2)
adamc@234 859
adamc@234 860 ]]
adamc@234 861
adam@386 862 Normal [eauto] still cannot prove the goal, so we eliminate the two new existential quantifiers. (Recall that [ex] is the underlying type family to which uses of the [exists] syntax are compiled.) *)
adamc@234 863
adamc@234 864 repeat match goal with
adamc@234 865 | [ H : ex _ |- _ ] => destruct H
adamc@234 866 end.
adamc@234 867
adamc@234 868 (** Now the goal is simple enough to solve by logic programming. *)
adamc@234 869
adamc@234 870 eauto.
adamc@234 871 Qed.
adamc@234 872 End t6.
adamc@234 873
adamc@234 874 (** 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@234 875
adamc@234 876 Section t7.
adamc@234 877 Variables A B : Type.
adamc@234 878 Variable Q : A -> Prop.
adamc@234 879 Variable P : A -> B -> Prop.
adamc@234 880 Variable f : A -> A -> A.
adamc@234 881 Variable g : B -> B -> B.
adamc@234 882
adamc@234 883 Hypothesis H1 : forall v, Q v -> exists u, P v u.
adamc@234 884 Hypothesis H2 : forall v1 u1 v2 u2,
adamc@234 885 P v1 u1
adamc@234 886 -> P v2 u2
adamc@234 887 -> P (f v1 v2) (g u1 u2).
adamc@234 888
adam@297 889 Theorem t7 : forall v1 v2, Q v1 -> Q v2 -> exists u1, exists u2, P (f v1 v2) (g u1 u2).
adamc@234 890 intros; do 2 insterKeep H1;
adamc@234 891 repeat match goal with
adamc@234 892 | [ H : ex _ |- _ ] => destruct H
adamc@234 893 end; eauto.
adamc@234 894
adamc@234 895 (** This proof script does not hit any errors until the very end, when an error message like this one is displayed.
adamc@234 896
adam@328 897 <<
adamc@234 898 No more subgoals but non-instantiated existential variables :
adamc@234 899 Existential 1 =
adam@328 900 >>
adam@445 901 %\vspace{-.35in}%[[
adamc@234 902 ?4384 : [A : Type
adamc@234 903 B : Type
adamc@234 904 Q : A -> Prop
adamc@234 905 P : A -> B -> Prop
adamc@234 906 f : A -> A -> A
adamc@234 907 g : B -> B -> B
adamc@234 908 H1 : forall v : A, Q v -> exists u : B, P v u
adamc@234 909 H2 : forall (v1 : A) (u1 : B) (v2 : A) (u2 : B),
adamc@234 910 P v1 u1 -> P v2 u2 -> P (f v1 v2) (g u1 u2)
adamc@234 911 v1 : A
adamc@234 912 v2 : A
adamc@234 913 H : Q v1
adamc@234 914 H0 : Q v2
adamc@234 915 H' : Q v2 -> exists u : B, P v2 u |- Q v2]
adamc@234 916 ]]
adamc@234 917
adam@431 918 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@234 919
adamc@234 920 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@234 921
adamc@234 922 Abort.
adamc@234 923 End t7.
adamc@234 924
adamc@234 925 Reset insterU.
adamc@234 926
adam@328 927 (** 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. Also recall that the tactic form %\index{tactics!solve}%[solve [ t ]] fails if [t] does not completely solve the goal. *)
adamc@234 928
adamc@234 929 Ltac insterU tac H :=
adamc@234 930 repeat match type of H with
adamc@234 931 | forall x : ?T, _ =>
adamc@234 932 match type of T with
adamc@234 933 | Prop =>
adamc@234 934 (let H' := fresh "H'" in
adam@328 935 assert (H' : T) by solve [ tac ];
adam@328 936 specialize (H H'); clear H')
adamc@234 937 || fail 1
adamc@234 938 | _ =>
adamc@234 939 let x := fresh "x" in
adamc@234 940 evar (x : T);
adam@328 941 let x' := eval unfold x in x in
adam@328 942 clear x; specialize (H x')
adamc@234 943 end
adamc@234 944 end.
adamc@234 945
adamc@234 946 Ltac insterKeep tac H :=
adamc@234 947 let H' := fresh "H'" in
adamc@234 948 generalize H; intro H'; insterU tac H'.
adamc@234 949
adamc@234 950 Section t7.
adamc@234 951 Variables A B : Type.
adamc@234 952 Variable Q : A -> Prop.
adamc@234 953 Variable P : A -> B -> Prop.
adamc@234 954 Variable f : A -> A -> A.
adamc@234 955 Variable g : B -> B -> B.
adamc@234 956
adamc@234 957 Hypothesis H1 : forall v, Q v -> exists u, P v u.
adamc@234 958 Hypothesis H2 : forall v1 u1 v2 u2,
adamc@234 959 P v1 u1
adamc@234 960 -> P v2 u2
adamc@234 961 -> P (f v1 v2) (g u1 u2).
adamc@234 962
adamc@234 963 Theorem t6 : forall v1 v2, Q v1 -> Q v2 -> exists u1, exists u2, P (f v1 v2) (g u1 u2).
adamc@234 964
adamc@234 965 (** 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@234 966
adamc@234 967 intros; do 2 insterKeep ltac:(idtac; match goal with
adamc@234 968 | [ H : Q ?v |- _ ] =>
adamc@234 969 match goal with
adamc@234 970 | [ _ : context[P v _] |- _ ] => fail 1
adamc@234 971 | _ => apply H
adamc@234 972 end
adamc@234 973 end) H1;
adamc@234 974 repeat match goal with
adamc@234 975 | [ H : ex _ |- _ ] => destruct H
adamc@234 976 end; eauto.
adamc@234 977 Qed.
adamc@234 978 End t7.
adamc@234 979
adamc@234 980 (** It is often useful to instantiate existential variables explicitly. A built-in tactic provides one way of doing so. *)
adamc@234 981
adamc@234 982 Theorem t8 : exists p : nat * nat, fst p = 3.
adamc@234 983 econstructor; instantiate (1 := (3, 2)); reflexivity.
adamc@234 984 Qed.
adamc@234 985
adam@460 986 (** 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@234 987
adam@328 988 The %\index{tactics!instantiate}%[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@234 989
adamc@234 990 Ltac equate x y :=
adam@460 991 let dummy := constr:(eq_refl x : x = y) in idtac.
adamc@234 992
adam@460 993 (** This tactic fails if it is not possible to prove [x = y] by [eq_refl]. We check the proof only for its unification side effects, ignoring the associated variable [dummy]. With [equate], we can build a less brittle version of the prior example. *)
adamc@234 994
adamc@234 995 Theorem t9 : exists p : nat * nat, fst p = 3.
adamc@234 996 econstructor; match goal with
adamc@234 997 | [ |- fst ?x = 3 ] => equate x (3, 2)
adamc@234 998 end; reflexivity.
adamc@234 999 Qed.
adam@386 1000
adam@386 1001 (** This technique is even more useful within recursive and iterative tactics that are meant to solve broad classes of goals. *)