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Pass through DataStruct, to incorporate new coqdoc features; globally replace [refl_equal] with [eq_refl]
author Adam Chlipala <adam@chlipala.net>
date Wed, 25 Jul 2012 18:10:26 -0400
parents 078edca127cf
children 85e743564b22
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@411 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@328 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@288 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
adamc@135 149 (** 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, that would mean that the whole pattern-match fails. In Coq, we backtrack and try the next pattern, which also matches. Its body tactic succeeds, so the overall tactic succeeds as well.
adamc@135 150
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@328 158 (** 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@328 183 (** 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 *)
adamc@135 270 (** [[
adamc@135 271 completer'.
adamc@220 272
adamc@205 273 ]]
adamc@205 274
adamc@135 275 Coq loops forever at this point. What went wrong? *)
adamc@220 276
adamc@135 277 Abort.
adamc@141 278 (* end thide *)
adamc@135 279 End firstorder'.
adamc@136 280
adamc@136 281 (** A few examples should illustrate the issue. Here we see a [match]-based proof that works fine: *)
adamc@136 282
adamc@136 283 Theorem t1 : forall x : nat, x = x.
adamc@136 284 match goal with
adamc@136 285 | [ |- forall x, _ ] => trivial
adamc@136 286 end.
adamc@141 287 (* begin thide *)
adamc@136 288 Qed.
adamc@141 289 (* end thide *)
adamc@136 290
adamc@136 291 (** This one fails. *)
adamc@136 292
adamc@141 293 (* begin thide *)
adamc@136 294 Theorem t1' : forall x : nat, x = x.
adamc@136 295 (** [[
adamc@136 296 match goal with
adamc@136 297 | [ |- forall x, ?P ] => trivial
adamc@136 298 end.
adam@328 299 ]]
adamc@136 300
adam@328 301 <<
adamc@136 302 User error: No matching clauses for match goal
adam@328 303 >>
adam@328 304 *)
adamc@220 305
adamc@136 306 Abort.
adamc@141 307 (* end thide *)
adamc@136 308
adam@411 309 (** 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@136 310
adam@386 311 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@136 312
adam@386 313 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@137 314
adamc@137 315
adamc@137 316 (** * Functional Programming in Ltac *)
adamc@137 317
adamc@141 318 (* EX: Write a list length function in Ltac. *)
adamc@141 319
adamc@137 320 (** 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@137 321
adamc@137 322 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@137 323
adamc@137 324 [[
adamc@137 325 Ltac length ls :=
adamc@137 326 match ls with
adamc@137 327 | nil => O
adamc@137 328 | _ :: ls' => S (length ls')
adamc@137 329 end.
adam@328 330 ]]
adamc@137 331
adam@328 332 <<
adamc@137 333 Error: The reference ls' was not found in the current environment
adam@328 334 >>
adamc@137 335
adamc@137 336 At this point, we hopefully remember that pattern variable names must be prefixed by question marks in Ltac.
adamc@137 337
adamc@137 338 [[
adamc@137 339 Ltac length ls :=
adamc@137 340 match ls with
adamc@137 341 | nil => O
adamc@137 342 | _ :: ?ls' => S (length ls')
adamc@137 343 end.
adamc@137 344 ]]
adamc@137 345
adam@328 346 <<
adam@328 347 Error: The reference S was not found in the current environment
adam@328 348 >>
adam@328 349
adam@411 350 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@137 351
adamc@141 352 (* begin thide *)
adamc@137 353 Ltac length ls :=
adamc@137 354 match ls with
adamc@137 355 | nil => O
adamc@137 356 | _ :: ?ls' => constr:(S (length ls'))
adamc@137 357 end.
adamc@137 358
adamc@137 359 (** This definition is accepted. It can be a little awkward to test Ltac definitions like this. Here is one method. *)
adamc@137 360
adamc@137 361 Goal False.
adamc@137 362 let n := length (1 :: 2 :: 3 :: nil) in
adamc@137 363 pose n.
adamc@137 364 (** [[
adamc@137 365 n := S (length (2 :: 3 :: nil)) : nat
adamc@137 366 ============================
adamc@137 367 False
adamc@220 368
adamc@137 369 ]]
adamc@137 370
adam@328 371 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@220 372
adam@328 373 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@220 374
adamc@220 375 Abort.
adamc@137 376
adamc@137 377 Reset length.
adamc@137 378
adamc@137 379 (** 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@137 380
adamc@137 381 Ltac length ls :=
adamc@137 382 match ls with
adamc@137 383 | nil => O
adamc@137 384 | _ :: ?ls' =>
adamc@137 385 let ls'' := length ls' in
adamc@137 386 constr:(S ls'')
adamc@137 387 end.
adamc@137 388
adamc@137 389 Goal False.
adamc@137 390 let n := length (1 :: 2 :: 3 :: nil) in
adamc@137 391 pose n.
adamc@137 392 (** [[
adamc@137 393 n := 3 : nat
adamc@137 394 ============================
adamc@137 395 False
adam@302 396 ]]
adam@302 397 *)
adamc@220 398
adamc@137 399 Abort.
adamc@141 400 (* end thide *)
adamc@141 401
adamc@141 402 (* EX: Write a list map function in Ltac. *)
adamc@137 403
adamc@137 404 (** We can also use anonymous function expressions and local function definitions in Ltac, as this example of a standard list [map] function shows. *)
adamc@137 405
adamc@141 406 (* begin thide *)
adamc@137 407 Ltac map T f :=
adamc@137 408 let rec map' ls :=
adamc@137 409 match ls with
adam@411 410 | nil => constr:(@nil T)
adamc@137 411 | ?x :: ?ls' =>
adamc@137 412 let x' := f x in
adamc@137 413 let ls'' := map' ls' in
adam@411 414 constr:(x' :: ls'')
adamc@137 415 end in
adamc@137 416 map'.
adamc@137 417
adam@411 418 (** 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@137 419
adam@288 420 Sometimes we need to employ the opposite direction of %``%#"#nonterminal escape,#"#%''% when we want to pass a complicated tactic expression as an argument to another tactic, as we might want to do in invoking [map]. *)
adamc@137 421
adamc@137 422 Goal False.
adam@411 423 let ls := map (nat * nat)%type ltac:(fun x => constr:(x, x)) (1 :: 2 :: 3 :: nil) in
adamc@137 424 pose ls.
adamc@137 425 (** [[
adamc@137 426 l := (1, 1) :: (2, 2) :: (3, 3) :: nil : list (nat * nat)
adamc@137 427 ============================
adamc@137 428 False
adam@302 429 ]]
adam@302 430 *)
adamc@220 431
adamc@137 432 Abort.
adamc@141 433 (* end thide *)
adamc@137 434
adam@398 435 (** 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 [map] 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@386 436
adam@386 437 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 [length] that prints a debug trace of the arguments it is called with. *)
adam@328 438
adam@334 439 (* begin thide *)
adam@328 440 Reset length.
adam@328 441
adam@328 442 Ltac length ls :=
adam@328 443 idtac ls;
adam@328 444 match ls with
adam@328 445 | nil => O
adam@328 446 | _ :: ?ls' =>
adam@328 447 let ls'' := length ls' in
adam@328 448 constr:(S ls'')
adam@328 449 end.
adam@328 450
adam@328 451 (** Coq accepts the tactic definition, but the code is fatally flawed and will always lead to dynamic type errors. *)
adam@328 452
adam@328 453 Goal False.
adam@328 454 (** %\vspace{-.15in}%[[
adam@328 455 let n := length (1 :: 2 :: 3 :: nil) in
adam@328 456 pose n.
adam@328 457 ]]
adam@328 458
adam@328 459 <<
adam@328 460 Error: variable n should be bound to a term.
adam@328 461 >> *)
adam@328 462 Abort.
adam@328 463
adam@398 464 (** 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@328 465
adam@328 466 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@328 467
adam@411 468 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@328 469
adam@328 470 Reset length.
adam@328 471
adam@328 472 Ltac length ls k :=
adam@328 473 idtac ls;
adam@328 474 match ls with
adam@328 475 | nil => k O
adam@328 476 | _ :: ?ls' => length ls' ltac:(fun n => k (S n))
adam@328 477 end.
adam@334 478 (* end thide *)
adam@328 479
adam@398 480 (** 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 [length]. The argument passed to [k] may be thought of as the return value of [length]. *)
adam@328 481
adam@334 482 (* begin thide *)
adam@328 483 Goal False.
adam@328 484 length (1 :: 2 :: 3 :: nil) ltac:(fun n => pose n).
adam@328 485 (** [[
adam@328 486 (1 :: 2 :: 3 :: nil)
adam@328 487 (2 :: 3 :: nil)
adam@328 488 (3 :: nil)
adam@328 489 nil
adam@328 490 ]]
adam@328 491 *)
adam@328 492 Abort.
adam@334 493 (* end thide *)
adam@328 494
adam@386 495 (** 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@386 496
adam@386 497 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@386 498
adam@398 499 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@328 500
adamc@138 501
adamc@139 502 (** * Recursive Proof Search *)
adamc@139 503
adamc@139 504 (** 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@139 505
adam@288 506 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@139 507
adamc@141 508 (* begin thide *)
adamc@139 509 Ltac inster n :=
adamc@139 510 intuition;
adamc@139 511 match n with
adamc@139 512 | S ?n' =>
adamc@139 513 match goal with
adamc@139 514 | [ H : forall x : ?T, _, x : ?T |- _ ] => generalize (H x); inster n'
adamc@139 515 end
adamc@139 516 end.
adamc@141 517 (* end thide *)
adamc@139 518
adam@386 519 (** 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@139 520
adamc@139 521 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@139 522
adamc@139 523 Section test_inster.
adamc@139 524 Variable A : Set.
adamc@139 525 Variables P Q : A -> Prop.
adamc@139 526 Variable f : A -> A.
adamc@139 527 Variable g : A -> A -> A.
adamc@139 528
adamc@139 529 Hypothesis H1 : forall x y, P (g x y) -> Q (f x).
adamc@139 530
adam@328 531 Theorem test_inster : forall x, P (g x x) -> Q (f x).
adamc@220 532 inster 2.
adamc@139 533 Qed.
adamc@139 534
adamc@139 535 Hypothesis H3 : forall u v, P u /\ P v /\ u <> v -> P (g u v).
adamc@139 536 Hypothesis H4 : forall u, Q (f u) -> P u /\ P (f u).
adamc@139 537
adamc@139 538 Theorem test_inster2 : forall x y, x <> y -> P x -> Q (f y) -> Q (f x).
adamc@220 539 inster 3.
adamc@139 540 Qed.
adamc@139 541 End test_inster.
adamc@139 542
adam@386 543 (** 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@140 544
adam@288 545 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@140 546
adam@328 547 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@140 548
adam@288 549 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@138 550
adamc@138 551 Definition imp (P1 P2 : Prop) := P1 -> P2.
adamc@140 552 Infix "-->" := imp (no associativity, at level 95).
adamc@140 553 Ltac imp := unfold imp; firstorder.
adamc@138 554
adamc@140 555 (** These lemmas about [imp] will be useful in the tactic that we will write. *)
adamc@138 556
adamc@138 557 Theorem and_True_prem : forall P Q,
adamc@138 558 (P /\ True --> Q)
adamc@138 559 -> (P --> Q).
adamc@138 560 imp.
adamc@138 561 Qed.
adamc@138 562
adamc@138 563 Theorem and_True_conc : forall P Q,
adamc@138 564 (P --> Q /\ True)
adamc@138 565 -> (P --> Q).
adamc@138 566 imp.
adamc@138 567 Qed.
adamc@138 568
adamc@138 569 Theorem assoc_prem1 : forall P Q R S,
adamc@138 570 (P /\ (Q /\ R) --> S)
adamc@138 571 -> ((P /\ Q) /\ R --> S).
adamc@138 572 imp.
adamc@138 573 Qed.
adamc@138 574
adamc@138 575 Theorem assoc_prem2 : forall P Q R S,
adamc@138 576 (Q /\ (P /\ R) --> S)
adamc@138 577 -> ((P /\ Q) /\ R --> S).
adamc@138 578 imp.
adamc@138 579 Qed.
adamc@138 580
adamc@138 581 Theorem comm_prem : forall P Q R,
adamc@138 582 (P /\ Q --> R)
adamc@138 583 -> (Q /\ P --> R).
adamc@138 584 imp.
adamc@138 585 Qed.
adamc@138 586
adamc@138 587 Theorem assoc_conc1 : forall P Q R S,
adamc@138 588 (S --> P /\ (Q /\ R))
adamc@138 589 -> (S --> (P /\ Q) /\ R).
adamc@138 590 imp.
adamc@138 591 Qed.
adamc@138 592
adamc@138 593 Theorem assoc_conc2 : forall P Q R S,
adamc@138 594 (S --> Q /\ (P /\ R))
adamc@138 595 -> (S --> (P /\ Q) /\ R).
adamc@138 596 imp.
adamc@138 597 Qed.
adamc@138 598
adamc@138 599 Theorem comm_conc : forall P Q R,
adamc@138 600 (R --> P /\ Q)
adamc@138 601 -> (R --> Q /\ P).
adamc@138 602 imp.
adamc@138 603 Qed.
adamc@138 604
adam@288 605 (** 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@140 606
adamc@138 607 Ltac search_prem tac :=
adamc@138 608 let rec search P :=
adamc@138 609 tac
adamc@138 610 || (apply and_True_prem; tac)
adamc@138 611 || match P with
adamc@138 612 | ?P1 /\ ?P2 =>
adamc@138 613 (apply assoc_prem1; search P1)
adamc@138 614 || (apply assoc_prem2; search P2)
adamc@138 615 end
adamc@138 616 in match goal with
adamc@138 617 | [ |- ?P /\ _ --> _ ] => search P
adamc@138 618 | [ |- _ /\ ?P --> _ ] => apply comm_prem; search P
adamc@138 619 | [ |- _ --> _ ] => progress (tac || (apply and_True_prem; tac))
adamc@138 620 end.
adamc@138 621
adam@328 622 (** 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.
adamc@140 623
adam@328 624 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@140 625
adamc@140 626 We will also want a dual function [search_conc], which does tree search through an [imp] conclusion. *)
adamc@140 627
adamc@138 628 Ltac search_conc tac :=
adamc@138 629 let rec search P :=
adamc@138 630 tac
adamc@138 631 || (apply and_True_conc; tac)
adamc@138 632 || match P with
adamc@138 633 | ?P1 /\ ?P2 =>
adamc@138 634 (apply assoc_conc1; search P1)
adamc@138 635 || (apply assoc_conc2; search P2)
adamc@138 636 end
adamc@138 637 in match goal with
adamc@138 638 | [ |- _ --> ?P /\ _ ] => search P
adamc@138 639 | [ |- _ --> _ /\ ?P ] => apply comm_conc; search P
adamc@138 640 | [ |- _ --> _ ] => progress (tac || (apply and_True_conc; tac))
adamc@138 641 end.
adamc@138 642
adamc@140 643 (** 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@140 644
adam@328 645 (* begin thide *)
adamc@138 646 Theorem False_prem : forall P Q,
adamc@138 647 False /\ P --> Q.
adamc@138 648 imp.
adamc@138 649 Qed.
adamc@138 650
adamc@138 651 Theorem True_conc : forall P Q : Prop,
adamc@138 652 (P --> Q)
adamc@138 653 -> (P --> True /\ Q).
adamc@138 654 imp.
adamc@138 655 Qed.
adamc@138 656
adamc@138 657 Theorem Match : forall P Q R : Prop,
adamc@138 658 (Q --> R)
adamc@138 659 -> (P /\ Q --> P /\ R).
adamc@138 660 imp.
adamc@138 661 Qed.
adamc@138 662
adamc@138 663 Theorem ex_prem : forall (T : Type) (P : T -> Prop) (Q R : Prop),
adamc@138 664 (forall x, P x /\ Q --> R)
adamc@138 665 -> (ex P /\ Q --> R).
adamc@138 666 imp.
adamc@138 667 Qed.
adamc@138 668
adamc@138 669 Theorem ex_conc : forall (T : Type) (P : T -> Prop) (Q R : Prop) x,
adamc@138 670 (Q --> P x /\ R)
adamc@138 671 -> (Q --> ex P /\ R).
adamc@138 672 imp.
adamc@138 673 Qed.
adamc@138 674
adam@288 675 (** We will also want a %``%#"#base case#"#%''% lemma for finishing proofs where cancelation has removed every constituent of the conclusion. *)
adamc@140 676
adamc@138 677 Theorem imp_True : forall P,
adamc@138 678 P --> True.
adamc@138 679 imp.
adamc@138 680 Qed.
adamc@138 681
adam@386 682 (** 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@140 683
adamc@138 684 Ltac matcher :=
adamc@138 685 intros;
adam@411 686 repeat search_prem ltac:(simple apply False_prem || (simple apply ex_prem; intro));
adam@411 687 repeat search_conc ltac:(simple apply True_conc || simple eapply ex_conc
adam@411 688 || search_prem ltac:(simple apply Match));
adamc@204 689 try simple apply imp_True.
adamc@141 690 (* end thide *)
adamc@140 691
adamc@140 692 (** Our tactic succeeds at proving a simple example. *)
adamc@138 693
adamc@138 694 Theorem t2 : forall P Q : Prop,
adamc@138 695 Q /\ (P /\ False) /\ P --> P /\ Q.
adamc@138 696 matcher.
adamc@138 697 Qed.
adamc@138 698
adamc@140 699 (** In the generated proof, we find a trace of the workings of the search tactics. *)
adamc@140 700
adamc@140 701 Print t2.
adamc@220 702 (** %\vspace{-.15in}% [[
adamc@140 703 t2 =
adamc@140 704 fun P Q : Prop =>
adamc@140 705 comm_prem (assoc_prem1 (assoc_prem2 (False_prem (P:=P /\ P /\ Q) (P /\ Q))))
adamc@140 706 : forall P Q : Prop, Q /\ (P /\ False) /\ P --> P /\ Q
adamc@220 707
adamc@220 708 ]]
adamc@140 709
adamc@220 710 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@328 754 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 instantiantiations are; rather, we want these choices filled with placeholders. We hope that, when we apply the specialized hypothesis later, syntactic unification will determine concrete values.
adamc@234 755
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@328 886 [[
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 ]]
adamc@234 903
adam@288 904 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@234 905
adamc@234 906 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@234 907
adamc@234 908 Abort.
adamc@234 909 End t7.
adamc@234 910
adamc@234 911 Reset insterU.
adamc@234 912
adam@328 913 (** 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@234 914
adamc@234 915 Ltac insterU tac H :=
adamc@234 916 repeat match type of H with
adamc@234 917 | forall x : ?T, _ =>
adamc@234 918 match type of T with
adamc@234 919 | Prop =>
adamc@234 920 (let H' := fresh "H'" in
adam@328 921 assert (H' : T) by solve [ tac ];
adam@328 922 specialize (H H'); clear H')
adamc@234 923 || fail 1
adamc@234 924 | _ =>
adamc@234 925 let x := fresh "x" in
adamc@234 926 evar (x : T);
adam@328 927 let x' := eval unfold x in x in
adam@328 928 clear x; specialize (H x')
adamc@234 929 end
adamc@234 930 end.
adamc@234 931
adamc@234 932 Ltac insterKeep tac H :=
adamc@234 933 let H' := fresh "H'" in
adamc@234 934 generalize H; intro H'; insterU tac H'.
adamc@234 935
adamc@234 936 Section t7.
adamc@234 937 Variables A B : Type.
adamc@234 938 Variable Q : A -> Prop.
adamc@234 939 Variable P : A -> B -> Prop.
adamc@234 940 Variable f : A -> A -> A.
adamc@234 941 Variable g : B -> B -> B.
adamc@234 942
adamc@234 943 Hypothesis H1 : forall v, Q v -> exists u, P v u.
adamc@234 944 Hypothesis H2 : forall v1 u1 v2 u2,
adamc@234 945 P v1 u1
adamc@234 946 -> P v2 u2
adamc@234 947 -> P (f v1 v2) (g u1 u2).
adamc@234 948
adamc@234 949 Theorem t6 : forall v1 v2, Q v1 -> Q v2 -> exists u1, exists u2, P (f v1 v2) (g u1 u2).
adamc@234 950
adamc@234 951 (** 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@234 952
adamc@234 953 intros; do 2 insterKeep ltac:(idtac; match goal with
adamc@234 954 | [ H : Q ?v |- _ ] =>
adamc@234 955 match goal with
adamc@234 956 | [ _ : context[P v _] |- _ ] => fail 1
adamc@234 957 | _ => apply H
adamc@234 958 end
adamc@234 959 end) H1;
adamc@234 960 repeat match goal with
adamc@234 961 | [ H : ex _ |- _ ] => destruct H
adamc@234 962 end; eauto.
adamc@234 963 Qed.
adamc@234 964 End t7.
adamc@234 965
adamc@234 966 (** It is often useful to instantiate existential variables explicitly. A built-in tactic provides one way of doing so. *)
adamc@234 967
adamc@234 968 Theorem t8 : exists p : nat * nat, fst p = 3.
adamc@234 969 econstructor; instantiate (1 := (3, 2)); reflexivity.
adamc@234 970 Qed.
adamc@234 971
adamc@234 972 (** 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@234 973
adam@328 974 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@234 975
adamc@234 976 Ltac equate x y :=
adamc@234 977 let H := fresh "H" in
adam@328 978 assert (H : x = y) by reflexivity; clear H.
adamc@234 979
adam@328 980 (** This tactic fails if it is not possible to prove [x = y] by [reflexivity]. We perform the proof only for its unification side effects, clearing the fact [x = y] afterward. With [equate], we can build a less brittle version of the prior example. *)
adamc@234 981
adamc@234 982 Theorem t9 : exists p : nat * nat, fst p = 3.
adamc@234 983 econstructor; match goal with
adamc@234 984 | [ |- fst ?x = 3 ] => equate x (3, 2)
adamc@234 985 end; reflexivity.
adamc@234 986 Qed.
adam@386 987
adam@386 988 (** This technique is even more useful within recursive and iterative tactics that are meant to solve broad classes of goals. *)