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author Adam Chlipala <adam@chlipala.net>
date Mon, 08 Oct 2012 16:04:49 -0400
parents 4320c1a967c2
children 1fd4109f7b31
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@411 120 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). *)
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
adamc@135 199 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@135 200
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
adamc@135 217 We can check that [completer] is working properly: *)
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
adamc@135 226 Theorem fo : forall x, P x -> S x.
adamc@141 227 (* begin thide *)
adamc@135 228 completer.
adamc@135 229 (** [[
adamc@135 230 x : A
adamc@135 231 H : P x
adamc@135 232 H0 : Q x
adamc@135 233 H3 : R x
adamc@135 234 H4 : S x
adamc@135 235 ============================
adamc@135 236 S x
adam@302 237 ]]
adam@302 238 *)
adamc@135 239
adamc@135 240 assumption.
adamc@135 241 Qed.
adamc@141 242 (* end thide *)
adamc@135 243 End firstorder.
adamc@135 244
adamc@135 245 (** 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@135 246
adamc@141 247 (* begin thide *)
adamc@135 248 Ltac completer' :=
adamc@135 249 repeat match goal with
adamc@135 250 | [ |- _ /\ _ ] => constructor
adamc@135 251 | [ H : _ /\ _ |- _ ] => destruct H
adam@328 252 | [ H : ?P -> _, H' : ?P |- _ ] => specialize (H H')
adamc@135 253 | [ |- forall x, _ ] => intro
adamc@135 254
adam@328 255 | [ H : forall x, ?P x -> _, H' : ?P ?X |- _ ] => extend (H X H')
adamc@135 256 end.
adamc@141 257 (* end thide *)
adamc@135 258
adam@411 259 (** 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@135 260
adamc@135 261 Section firstorder'.
adamc@135 262 Variable A : Set.
adamc@135 263 Variables P Q R S : A -> Prop.
adamc@135 264
adamc@135 265 Hypothesis H1 : forall x, P x -> Q x /\ R x.
adamc@135 266 Hypothesis H2 : forall x, R x -> S x.
adamc@135 267
adamc@135 268 Theorem fo' : forall x, P x -> S x.
adamc@141 269 (* begin thide *)
adam@445 270 (** %\vspace{-.25in}%[[
adamc@135 271 completer'.
adamc@205 272 ]]
adam@445 273 %\vspace{-.15in}%Coq loops forever at this point. What went wrong? *)
adamc@220 274
adamc@135 275 Abort.
adamc@141 276 (* end thide *)
adamc@135 277 End firstorder'.
adamc@136 278
adamc@136 279 (** A few examples should illustrate the issue. Here we see a [match]-based proof that works fine: *)
adamc@136 280
adamc@136 281 Theorem t1 : forall x : nat, x = x.
adamc@136 282 match goal with
adamc@136 283 | [ |- forall x, _ ] => trivial
adamc@136 284 end.
adamc@141 285 (* begin thide *)
adamc@136 286 Qed.
adamc@141 287 (* end thide *)
adamc@136 288
adamc@136 289 (** This one fails. *)
adamc@136 290
adamc@141 291 (* begin thide *)
adamc@136 292 Theorem t1' : forall x : nat, x = x.
adam@445 293 (** %\vspace{-.25in}%[[
adamc@136 294 match goal with
adamc@136 295 | [ |- forall x, ?P ] => trivial
adamc@136 296 end.
adam@328 297 ]]
adamc@136 298
adam@328 299 <<
adamc@136 300 User error: No matching clauses for match goal
adam@328 301 >>
adam@328 302 *)
adamc@220 303
adamc@136 304 Abort.
adamc@141 305 (* end thide *)
adamc@136 306
adam@411 307 (** The problem is that unification variables may not contain locally bound variables. In this case, [?P] would need to be bound to [x = x], which contains the local quantified variable [x]. By using a wildcard in the earlier version, we avoided this restriction. To understand why this applies to the [completer] tactics, recall that, in Coq, implication is shorthand for degenerate universal quantification where the quantified variable is not used. Nonetheless, in an Ltac pattern, Coq is happy to match a wildcard implication against a universal quantification.
adamc@136 308
adam@431 309 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 the next chapter.
adamc@136 310
adam@431 311 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 infinite-looping behavior of [completer']. We unintentionally match quantified facts with the modus ponens rule, circumventing the "already present" check and leading to different behavior, where the same fact may be added to the context repeatedly in an infinite loop. Our earlier [completer] tactic uses a modus ponens rule that matches the implication conclusion with a variable, which blocks matching against non-trivial universal quantifiers. *)
adamc@137 312
adamc@137 313
adamc@137 314 (** * Functional Programming in Ltac *)
adamc@137 315
adamc@141 316 (* EX: Write a list length function in Ltac. *)
adamc@141 317
adamc@137 318 (** 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@137 319
adamc@137 320 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@137 321 [[
adamc@137 322 Ltac length ls :=
adamc@137 323 match ls with
adamc@137 324 | nil => O
adamc@137 325 | _ :: ls' => S (length ls')
adamc@137 326 end.
adam@328 327 ]]
adamc@137 328
adam@328 329 <<
adamc@137 330 Error: The reference ls' was not found in the current environment
adam@328 331 >>
adamc@137 332
adamc@137 333 At this point, we hopefully remember that pattern variable names must be prefixed by question marks in 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.
adamc@137 340 ]]
adamc@137 341
adam@328 342 <<
adam@328 343 Error: The reference S was not found in the current environment
adam@328 344 >>
adam@328 345
adam@431 346 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@137 347
adamc@141 348 (* begin thide *)
adamc@137 349 Ltac length ls :=
adamc@137 350 match ls with
adamc@137 351 | nil => O
adamc@137 352 | _ :: ?ls' => constr:(S (length ls'))
adamc@137 353 end.
adamc@137 354
adam@445 355 (** This definition is accepted. It can be a little awkward to test Ltac definitions like this one. Here is one method. *)
adamc@137 356
adamc@137 357 Goal False.
adamc@137 358 let n := length (1 :: 2 :: 3 :: nil) in
adamc@137 359 pose n.
adamc@137 360 (** [[
adamc@137 361 n := S (length (2 :: 3 :: nil)) : nat
adamc@137 362 ============================
adamc@137 363 False
adamc@137 364 ]]
adamc@137 365
adam@328 366 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@220 367
adam@328 368 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@220 369
adamc@220 370 Abort.
adamc@137 371
adamc@137 372 Reset length.
adamc@137 373
adamc@137 374 (** 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@137 375
adamc@137 376 Ltac length ls :=
adamc@137 377 match ls with
adamc@137 378 | nil => O
adamc@137 379 | _ :: ?ls' =>
adamc@137 380 let ls'' := length ls' in
adamc@137 381 constr:(S ls'')
adamc@137 382 end.
adamc@137 383
adamc@137 384 Goal False.
adamc@137 385 let n := length (1 :: 2 :: 3 :: nil) in
adamc@137 386 pose n.
adamc@137 387 (** [[
adamc@137 388 n := 3 : nat
adamc@137 389 ============================
adamc@137 390 False
adam@302 391 ]]
adam@302 392 *)
adamc@220 393
adamc@137 394 Abort.
adamc@141 395 (* end thide *)
adamc@141 396
adamc@141 397 (* EX: Write a list map function in Ltac. *)
adamc@137 398
adam@431 399 (* begin hide *)
adam@437 400 (* begin thide *)
adam@431 401 Definition mapp := (map, list).
adam@437 402 (* end thide *)
adam@431 403 (* end hide *)
adam@431 404
adamc@137 405 (** We can also use anonymous function expressions and local function definitions in Ltac, as this example of a standard list [map] function shows. *)
adamc@137 406
adamc@141 407 (* begin thide *)
adamc@137 408 Ltac map T f :=
adamc@137 409 let rec map' ls :=
adamc@137 410 match ls with
adam@411 411 | nil => constr:(@nil T)
adamc@137 412 | ?x :: ?ls' =>
adamc@137 413 let x' := f x in
adamc@137 414 let ls'' := map' ls' in
adam@411 415 constr:(x' :: ls'')
adamc@137 416 end in
adamc@137 417 map'.
adamc@137 418
adam@411 419 (** 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@137 420
adam@431 421 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@137 422
adamc@137 423 Goal False.
adam@411 424 let ls := map (nat * nat)%type ltac:(fun x => constr:(x, x)) (1 :: 2 :: 3 :: nil) in
adamc@137 425 pose ls.
adamc@137 426 (** [[
adamc@137 427 l := (1, 1) :: (2, 2) :: (3, 3) :: nil : list (nat * nat)
adamc@137 428 ============================
adamc@137 429 False
adam@302 430 ]]
adam@302 431 *)
adamc@220 432
adamc@137 433 Abort.
adamc@141 434 (* end thide *)
adamc@137 435
adam@431 436 (** 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@386 437
adam@431 438 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@328 439
adam@334 440 (* begin thide *)
adam@328 441 Reset length.
adam@328 442
adam@328 443 Ltac length ls :=
adam@328 444 idtac ls;
adam@328 445 match ls with
adam@328 446 | nil => O
adam@328 447 | _ :: ?ls' =>
adam@328 448 let ls'' := length ls' in
adam@328 449 constr:(S ls'')
adam@328 450 end.
adam@328 451
adam@328 452 (** Coq accepts the tactic definition, but the code is fatally flawed and will always lead to dynamic type errors. *)
adam@328 453
adam@328 454 Goal False.
adam@328 455 (** %\vspace{-.15in}%[[
adam@328 456 let n := length (1 :: 2 :: 3 :: nil) in
adam@328 457 pose n.
adam@328 458 ]]
adam@328 459
adam@328 460 <<
adam@328 461 Error: variable n should be bound to a term.
adam@328 462 >> *)
adam@328 463 Abort.
adam@328 464
adam@431 465 (** 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. Side-effecting Haskell programs 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@328 466
adam@328 467 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@328 468
adam@431 469 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@328 470
adam@328 471 Reset length.
adam@328 472
adam@328 473 Ltac length ls k :=
adam@328 474 idtac ls;
adam@328 475 match ls with
adam@328 476 | nil => k O
adam@328 477 | _ :: ?ls' => length ls' ltac:(fun n => k (S n))
adam@328 478 end.
adam@334 479 (* end thide *)
adam@328 480
adam@431 481 (** 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@328 482
adam@334 483 (* begin thide *)
adam@328 484 Goal False.
adam@328 485 length (1 :: 2 :: 3 :: nil) ltac:(fun n => pose n).
adam@328 486 (** [[
adam@328 487 (1 :: 2 :: 3 :: nil)
adam@328 488 (2 :: 3 :: nil)
adam@328 489 (3 :: nil)
adam@328 490 nil
adam@328 491 ]]
adam@328 492 *)
adam@328 493 Abort.
adam@334 494 (* end thide *)
adam@328 495
adam@386 496 (** 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@386 497
adam@431 498 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@386 499
adam@398 500 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@328 501
adamc@138 502
adamc@139 503 (** * Recursive Proof Search *)
adamc@139 504
adamc@139 505 (** 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@139 506
adam@431 507 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@139 508
adamc@141 509 (* begin thide *)
adamc@139 510 Ltac inster n :=
adamc@139 511 intuition;
adamc@139 512 match n with
adamc@139 513 | S ?n' =>
adamc@139 514 match goal with
adam@460 515 | [ H : forall x : ?T, _, y : ?T |- _ ] => generalize (H y); inster n'
adamc@139 516 end
adamc@139 517 end.
adamc@141 518 (* end thide *)
adamc@139 519
adam@386 520 (** The tactic begins by applying propositional simplification. Next, it checks if any chain length remains, failing if not. If so, it tries all possible ways of instantiating quantified hypotheses with properly typed local variables. It is critical to realize that, if the recursive call [inster n'] fails, then the [match goal] just seeks out another way of unifying its pattern against proof state. Thus, this small amount of code provides an elegant demonstration of how backtracking [match] enables exhaustive search.
adamc@139 521
adamc@139 522 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@139 523
adamc@139 524 Section test_inster.
adamc@139 525 Variable A : Set.
adamc@139 526 Variables P Q : A -> Prop.
adamc@139 527 Variable f : A -> A.
adamc@139 528 Variable g : A -> A -> A.
adamc@139 529
adamc@139 530 Hypothesis H1 : forall x y, P (g x y) -> Q (f x).
adamc@139 531
adam@328 532 Theorem test_inster : forall x, P (g x x) -> Q (f x).
adamc@220 533 inster 2.
adamc@139 534 Qed.
adamc@139 535
adamc@139 536 Hypothesis H3 : forall u v, P u /\ P v /\ u <> v -> P (g u v).
adamc@139 537 Hypothesis H4 : forall u, Q (f u) -> P u /\ P (f u).
adamc@139 538
adamc@139 539 Theorem test_inster2 : forall x y, x <> y -> P x -> Q (f y) -> Q (f x).
adamc@220 540 inster 3.
adamc@139 541 Qed.
adamc@139 542 End test_inster.
adamc@139 543
adam@431 544 (** 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@140 545
adam@431 546 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@140 547
adam@431 548 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@140 549
adam@431 550 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@138 551
adamc@138 552 Definition imp (P1 P2 : Prop) := P1 -> P2.
adamc@140 553 Infix "-->" := imp (no associativity, at level 95).
adamc@140 554 Ltac imp := unfold imp; firstorder.
adamc@138 555
adamc@140 556 (** These lemmas about [imp] will be useful in the tactic that we will write. *)
adamc@138 557
adamc@138 558 Theorem and_True_prem : forall P Q,
adamc@138 559 (P /\ True --> Q)
adamc@138 560 -> (P --> Q).
adamc@138 561 imp.
adamc@138 562 Qed.
adamc@138 563
adamc@138 564 Theorem and_True_conc : forall P Q,
adamc@138 565 (P --> Q /\ True)
adamc@138 566 -> (P --> Q).
adamc@138 567 imp.
adamc@138 568 Qed.
adamc@138 569
adamc@138 570 Theorem assoc_prem1 : forall P Q R S,
adamc@138 571 (P /\ (Q /\ R) --> S)
adamc@138 572 -> ((P /\ Q) /\ R --> S).
adamc@138 573 imp.
adamc@138 574 Qed.
adamc@138 575
adamc@138 576 Theorem assoc_prem2 : forall P Q R S,
adamc@138 577 (Q /\ (P /\ R) --> S)
adamc@138 578 -> ((P /\ Q) /\ R --> S).
adamc@138 579 imp.
adamc@138 580 Qed.
adamc@138 581
adamc@138 582 Theorem comm_prem : forall P Q R,
adamc@138 583 (P /\ Q --> R)
adamc@138 584 -> (Q /\ P --> R).
adamc@138 585 imp.
adamc@138 586 Qed.
adamc@138 587
adamc@138 588 Theorem assoc_conc1 : forall P Q R S,
adamc@138 589 (S --> P /\ (Q /\ R))
adamc@138 590 -> (S --> (P /\ Q) /\ R).
adamc@138 591 imp.
adamc@138 592 Qed.
adamc@138 593
adamc@138 594 Theorem assoc_conc2 : forall P Q R S,
adamc@138 595 (S --> Q /\ (P /\ R))
adamc@138 596 -> (S --> (P /\ Q) /\ R).
adamc@138 597 imp.
adamc@138 598 Qed.
adamc@138 599
adamc@138 600 Theorem comm_conc : forall P Q R,
adamc@138 601 (R --> P /\ Q)
adamc@138 602 -> (R --> Q /\ P).
adamc@138 603 imp.
adamc@138 604 Qed.
adamc@138 605
adam@431 606 (** 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@140 607
adamc@138 608 Ltac search_prem tac :=
adamc@138 609 let rec search P :=
adamc@138 610 tac
adamc@138 611 || (apply and_True_prem; tac)
adamc@138 612 || match P with
adamc@138 613 | ?P1 /\ ?P2 =>
adamc@138 614 (apply assoc_prem1; search P1)
adamc@138 615 || (apply assoc_prem2; search P2)
adamc@138 616 end
adamc@138 617 in match goal with
adamc@138 618 | [ |- ?P /\ _ --> _ ] => search P
adamc@138 619 | [ |- _ /\ ?P --> _ ] => apply comm_prem; search P
adamc@138 620 | [ |- _ --> _ ] => progress (tac || (apply and_True_prem; tac))
adamc@138 621 end.
adamc@138 622
adam@460 623 (** 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@140 624
adam@328 625 The [search] function itself tries the same tricks as in the last case of the final [match]. Additionally, if neither works, it checks if [P] is a conjunction. If so, it calls itself recursively on each conjunct, first applying associativity lemmas to maintain the goal-form invariant.
adamc@140 626
adamc@140 627 We will also want a dual function [search_conc], which does tree search through an [imp] conclusion. *)
adamc@140 628
adamc@138 629 Ltac search_conc tac :=
adamc@138 630 let rec search P :=
adamc@138 631 tac
adamc@138 632 || (apply and_True_conc; tac)
adamc@138 633 || match P with
adamc@138 634 | ?P1 /\ ?P2 =>
adamc@138 635 (apply assoc_conc1; search P1)
adamc@138 636 || (apply assoc_conc2; search P2)
adamc@138 637 end
adamc@138 638 in match goal with
adamc@138 639 | [ |- _ --> ?P /\ _ ] => search P
adamc@138 640 | [ |- _ --> _ /\ ?P ] => apply comm_conc; search P
adamc@138 641 | [ |- _ --> _ ] => progress (tac || (apply and_True_conc; tac))
adamc@138 642 end.
adamc@138 643
adamc@140 644 (** 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@140 645
adam@328 646 (* begin thide *)
adamc@138 647 Theorem False_prem : forall P Q,
adamc@138 648 False /\ P --> Q.
adamc@138 649 imp.
adamc@138 650 Qed.
adamc@138 651
adamc@138 652 Theorem True_conc : forall P Q : Prop,
adamc@138 653 (P --> Q)
adamc@138 654 -> (P --> True /\ Q).
adamc@138 655 imp.
adamc@138 656 Qed.
adamc@138 657
adamc@138 658 Theorem Match : forall P Q R : Prop,
adamc@138 659 (Q --> R)
adamc@138 660 -> (P /\ Q --> P /\ R).
adamc@138 661 imp.
adamc@138 662 Qed.
adamc@138 663
adamc@138 664 Theorem ex_prem : forall (T : Type) (P : T -> Prop) (Q R : Prop),
adamc@138 665 (forall x, P x /\ Q --> R)
adamc@138 666 -> (ex P /\ Q --> R).
adamc@138 667 imp.
adamc@138 668 Qed.
adamc@138 669
adamc@138 670 Theorem ex_conc : forall (T : Type) (P : T -> Prop) (Q R : Prop) x,
adamc@138 671 (Q --> P x /\ R)
adamc@138 672 -> (Q --> ex P /\ R).
adamc@138 673 imp.
adamc@138 674 Qed.
adamc@138 675
adam@465 676 (** We will also want a "base case" lemma for finishing proofs where cancellation has removed every constituent of the conclusion. *)
adamc@140 677
adamc@138 678 Theorem imp_True : forall P,
adamc@138 679 P --> True.
adamc@138 680 imp.
adamc@138 681 Qed.
adamc@138 682
adam@386 683 (** 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@140 684
adamc@138 685 Ltac matcher :=
adamc@138 686 intros;
adam@411 687 repeat search_prem ltac:(simple apply False_prem || (simple apply ex_prem; intro));
adam@411 688 repeat search_conc ltac:(simple apply True_conc || simple eapply ex_conc
adam@411 689 || search_prem ltac:(simple apply Match));
adamc@204 690 try simple apply imp_True.
adamc@141 691 (* end thide *)
adamc@140 692
adamc@140 693 (** Our tactic succeeds at proving a simple example. *)
adamc@138 694
adamc@138 695 Theorem t2 : forall P Q : Prop,
adamc@138 696 Q /\ (P /\ False) /\ P --> P /\ Q.
adamc@138 697 matcher.
adamc@138 698 Qed.
adamc@138 699
adamc@140 700 (** In the generated proof, we find a trace of the workings of the search tactics. *)
adamc@140 701
adamc@140 702 Print t2.
adamc@220 703 (** %\vspace{-.15in}% [[
adamc@140 704 t2 =
adamc@140 705 fun P Q : Prop =>
adamc@140 706 comm_prem (assoc_prem1 (assoc_prem2 (False_prem (P:=P /\ P /\ Q) (P /\ Q))))
adamc@140 707 : forall P Q : Prop, Q /\ (P /\ False) /\ P --> P /\ Q
adamc@220 708 ]]
adamc@140 709
adam@460 710 %\smallskip{}%We can also see that [matcher] is well-suited for cases where some human intervention is needed after the automation finishes. *)
adamc@140 711
adamc@138 712 Theorem t3 : forall P Q R : Prop,
adamc@138 713 P /\ Q --> Q /\ R /\ P.
adamc@138 714 matcher.
adamc@140 715 (** [[
adamc@140 716 ============================
adamc@140 717 True --> R
adamc@220 718
adamc@140 719 ]]
adamc@140 720
adam@328 721 Our tactic canceled those conjuncts that it was able to cancel, leaving a simplified subgoal for us, much as [intuition] does. *)
adamc@220 722
adamc@138 723 Abort.
adamc@138 724
adam@328 725 (** 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@140 726
adamc@138 727 Theorem t4 : forall (P : nat -> Prop) Q, (exists x, P x /\ Q) --> Q /\ (exists x, P x).
adamc@138 728 matcher.
adamc@138 729 Qed.
adamc@138 730
adamc@140 731 Print t4.
adamc@220 732 (** %\vspace{-.15in}% [[
adamc@140 733 t4 =
adamc@140 734 fun (P : nat -> Prop) (Q : Prop) =>
adamc@140 735 and_True_prem
adamc@140 736 (ex_prem (P:=fun x : nat => P x /\ Q)
adamc@140 737 (fun x : nat =>
adamc@140 738 assoc_prem2
adamc@140 739 (Match (P:=Q)
adamc@140 740 (and_True_conc
adamc@140 741 (ex_conc (fun x0 : nat => P x0) x
adamc@140 742 (Match (P:=P x) (imp_True (P:=True))))))))
adamc@140 743 : forall (P : nat -> Prop) (Q : Prop),
adamc@140 744 (exists x : nat, P x /\ Q) --> Q /\ (exists x : nat, P x)
adam@302 745 ]]
adam@386 746
adam@386 747 This proof term is a mouthful, and we can be glad that we did not build it manually! *)
adamc@234 748
adamc@234 749
adamc@234 750 (** * Creating Unification Variables *)
adamc@234 751
adam@398 752 (** 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@234 753
adam@465 754 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@234 755
adamc@234 756 Before we are ready to write a tactic, we can try out its ingredients one at a time. *)
adamc@234 757
adamc@234 758 Theorem t5 : (forall x : nat, S x > x) -> 2 > 1.
adamc@234 759 intros.
adamc@234 760
adamc@234 761 (** [[
adamc@234 762 H : forall x : nat, S x > x
adamc@234 763 ============================
adamc@234 764 2 > 1
adamc@234 765
adamc@234 766 ]]
adamc@234 767
adam@328 768 To instantiate [H] generically, we first need to name the value to be used for [x].%\index{tactics!evar}% *)
adamc@234 769
adamc@234 770 evar (y : nat).
adamc@234 771
adamc@234 772 (** [[
adamc@234 773 H : forall x : nat, S x > x
adamc@234 774 y := ?279 : nat
adamc@234 775 ============================
adamc@234 776 2 > 1
adamc@234 777
adamc@234 778 ]]
adamc@234 779
adam@328 780 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@234 781
adam@328 782 let y' := eval unfold y in y in
adam@386 783 clear y; specialize (H y').
adamc@234 784
adamc@234 785 (** [[
adam@386 786 H : S ?279 > ?279
adamc@234 787 ============================
adam@386 788 2 > 1
adamc@234 789
adamc@234 790 ]]
adamc@234 791
adam@386 792 Our instantiation was successful. We can finish the proof by using [apply]'s unification to figure out the proper value of [?279]. *)
adamc@234 793
adamc@234 794 apply H.
adamc@234 795 Qed.
adamc@234 796
adamc@234 797 (** Now we can write a tactic that encapsulates the pattern we just employed, instantiating all quantifiers of a particular hypothesis. *)
adamc@234 798
adamc@234 799 Ltac insterU H :=
adamc@234 800 repeat match type of H with
adamc@234 801 | forall x : ?T, _ =>
adamc@234 802 let x := fresh "x" in
adamc@234 803 evar (x : T);
adam@328 804 let x' := eval unfold x in x in
adam@328 805 clear x; specialize (H x')
adamc@234 806 end.
adamc@234 807
adamc@234 808 Theorem t5' : (forall x : nat, S x > x) -> 2 > 1.
adamc@234 809 intro H; insterU H; apply H.
adamc@234 810 Qed.
adamc@234 811
adam@328 812 (** 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@234 813
adamc@234 814 Ltac insterKeep H :=
adamc@234 815 let H' := fresh "H'" in
adamc@234 816 generalize H; intro H'; insterU H'.
adamc@234 817
adamc@234 818 Section t6.
adamc@234 819 Variables A B : Type.
adamc@234 820 Variable P : A -> B -> Prop.
adamc@234 821 Variable f : A -> A -> A.
adamc@234 822 Variable g : B -> B -> B.
adamc@234 823
adamc@234 824 Hypothesis H1 : forall v, exists u, P v u.
adamc@234 825 Hypothesis H2 : forall v1 u1 v2 u2,
adamc@234 826 P v1 u1
adamc@234 827 -> P v2 u2
adamc@234 828 -> P (f v1 v2) (g u1 u2).
adamc@234 829
adamc@234 830 Theorem t6 : forall v1 v2, exists u1, exists u2, P (f v1 v2) (g u1 u2).
adamc@234 831 intros.
adamc@234 832
adam@328 833 (** 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@234 834
adamc@234 835 do 2 insterKeep H1.
adamc@234 836
adamc@234 837 (** Our proof state is extended with two generic instances of [H1].
adamc@234 838
adamc@234 839 [[
adamc@234 840 H' : exists u : B, P ?4289 u
adamc@234 841 H'0 : exists u : B, P ?4288 u
adamc@234 842 ============================
adamc@234 843 exists u1 : B, exists u2 : B, P (f v1 v2) (g u1 u2)
adamc@234 844
adamc@234 845 ]]
adamc@234 846
adam@386 847 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@234 848
adamc@234 849 repeat match goal with
adamc@234 850 | [ H : ex _ |- _ ] => destruct H
adamc@234 851 end.
adamc@234 852
adamc@234 853 (** Now the goal is simple enough to solve by logic programming. *)
adamc@234 854
adamc@234 855 eauto.
adamc@234 856 Qed.
adamc@234 857 End t6.
adamc@234 858
adamc@234 859 (** 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@234 860
adamc@234 861 Section t7.
adamc@234 862 Variables A B : Type.
adamc@234 863 Variable Q : A -> Prop.
adamc@234 864 Variable P : A -> B -> Prop.
adamc@234 865 Variable f : A -> A -> A.
adamc@234 866 Variable g : B -> B -> B.
adamc@234 867
adamc@234 868 Hypothesis H1 : forall v, Q v -> exists u, P v u.
adamc@234 869 Hypothesis H2 : forall v1 u1 v2 u2,
adamc@234 870 P v1 u1
adamc@234 871 -> P v2 u2
adamc@234 872 -> P (f v1 v2) (g u1 u2).
adamc@234 873
adam@297 874 Theorem t7 : forall v1 v2, Q v1 -> Q v2 -> exists u1, exists u2, P (f v1 v2) (g u1 u2).
adamc@234 875 intros; do 2 insterKeep H1;
adamc@234 876 repeat match goal with
adamc@234 877 | [ H : ex _ |- _ ] => destruct H
adamc@234 878 end; eauto.
adamc@234 879
adamc@234 880 (** This proof script does not hit any errors until the very end, when an error message like this one is displayed.
adamc@234 881
adam@328 882 <<
adamc@234 883 No more subgoals but non-instantiated existential variables :
adamc@234 884 Existential 1 =
adam@328 885 >>
adam@445 886 %\vspace{-.35in}%[[
adamc@234 887 ?4384 : [A : Type
adamc@234 888 B : Type
adamc@234 889 Q : A -> Prop
adamc@234 890 P : A -> B -> Prop
adamc@234 891 f : A -> A -> A
adamc@234 892 g : B -> B -> B
adamc@234 893 H1 : forall v : A, Q v -> exists u : B, P v u
adamc@234 894 H2 : forall (v1 : A) (u1 : B) (v2 : A) (u2 : B),
adamc@234 895 P v1 u1 -> P v2 u2 -> P (f v1 v2) (g u1 u2)
adamc@234 896 v1 : A
adamc@234 897 v2 : A
adamc@234 898 H : Q v1
adamc@234 899 H0 : Q v2
adamc@234 900 H' : Q v2 -> exists u : B, P v2 u |- Q v2]
adamc@234 901 ]]
adamc@234 902
adam@431 903 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@234 904
adamc@234 905 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@234 906
adamc@234 907 Abort.
adamc@234 908 End t7.
adamc@234 909
adamc@234 910 Reset insterU.
adamc@234 911
adam@328 912 (** 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@234 913
adamc@234 914 Ltac insterU tac H :=
adamc@234 915 repeat match type of H with
adamc@234 916 | forall x : ?T, _ =>
adamc@234 917 match type of T with
adamc@234 918 | Prop =>
adamc@234 919 (let H' := fresh "H'" in
adam@328 920 assert (H' : T) by solve [ tac ];
adam@328 921 specialize (H H'); clear H')
adamc@234 922 || fail 1
adamc@234 923 | _ =>
adamc@234 924 let x := fresh "x" in
adamc@234 925 evar (x : T);
adam@328 926 let x' := eval unfold x in x in
adam@328 927 clear x; specialize (H x')
adamc@234 928 end
adamc@234 929 end.
adamc@234 930
adamc@234 931 Ltac insterKeep tac H :=
adamc@234 932 let H' := fresh "H'" in
adamc@234 933 generalize H; intro H'; insterU tac H'.
adamc@234 934
adamc@234 935 Section t7.
adamc@234 936 Variables A B : Type.
adamc@234 937 Variable Q : A -> Prop.
adamc@234 938 Variable P : A -> B -> Prop.
adamc@234 939 Variable f : A -> A -> A.
adamc@234 940 Variable g : B -> B -> B.
adamc@234 941
adamc@234 942 Hypothesis H1 : forall v, Q v -> exists u, P v u.
adamc@234 943 Hypothesis H2 : forall v1 u1 v2 u2,
adamc@234 944 P v1 u1
adamc@234 945 -> P v2 u2
adamc@234 946 -> P (f v1 v2) (g u1 u2).
adamc@234 947
adamc@234 948 Theorem t6 : forall v1 v2, Q v1 -> Q v2 -> exists u1, exists u2, P (f v1 v2) (g u1 u2).
adamc@234 949
adamc@234 950 (** 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@234 951
adamc@234 952 intros; do 2 insterKeep ltac:(idtac; match goal with
adamc@234 953 | [ H : Q ?v |- _ ] =>
adamc@234 954 match goal with
adamc@234 955 | [ _ : context[P v _] |- _ ] => fail 1
adamc@234 956 | _ => apply H
adamc@234 957 end
adamc@234 958 end) H1;
adamc@234 959 repeat match goal with
adamc@234 960 | [ H : ex _ |- _ ] => destruct H
adamc@234 961 end; eauto.
adamc@234 962 Qed.
adamc@234 963 End t7.
adamc@234 964
adamc@234 965 (** It is often useful to instantiate existential variables explicitly. A built-in tactic provides one way of doing so. *)
adamc@234 966
adamc@234 967 Theorem t8 : exists p : nat * nat, fst p = 3.
adamc@234 968 econstructor; instantiate (1 := (3, 2)); reflexivity.
adamc@234 969 Qed.
adamc@234 970
adam@460 971 (** 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@234 972
adam@328 973 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@234 974
adamc@234 975 Ltac equate x y :=
adam@460 976 let dummy := constr:(eq_refl x : x = y) in idtac.
adamc@234 977
adam@460 978 (** 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@234 979
adamc@234 980 Theorem t9 : exists p : nat * nat, fst p = 3.
adamc@234 981 econstructor; match goal with
adamc@234 982 | [ |- fst ?x = 3 ] => equate x (3, 2)
adamc@234 983 end; reflexivity.
adamc@234 984 Qed.
adam@386 985
adam@386 986 (** This technique is even more useful within recursive and iterative tactics that are meant to solve broad classes of goals. *)