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LogicProg exercise on group theory
author Adam Chlipala <adam@chlipala.net>
date Thu, 22 Sep 2011 11:09:10 -0400
parents 06d11a6363cd
children cbeccef45f4e
rev   line source
adam@297 1 (* Copyright (c) 2008-2011, Adam Chlipala
adamc@132 2 *
adamc@132 3 * This work is licensed under a
adamc@132 4 * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
adamc@132 5 * Unported License.
adamc@132 6 * The license text is available at:
adamc@132 7 * http://creativecommons.org/licenses/by-nc-nd/3.0/
adamc@132 8 *)
adamc@132 9
adamc@132 10 (* begin hide *)
adamc@132 11 Require Import List.
adamc@132 12
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
adamc@132 21 (** We have seen many examples of proof automation so far. This chapter aims to give a principled presentation of the features of Ltac, focusing in particular on the Ltac [match] construct, which supports a novel approach to backtracking search. First, though, we will run through some useful automation tactics that are built into Coq. They are described in detail in the manual, so we only outline what is possible. *)
adamc@132 22
adamc@132 23 (** * Some Built-In Automation Tactics *)
adamc@132 24
adamc@132 25 (** A number of tactics are called repeatedly by [crush]. [intuition] simplifies propositional structure of goals. [congruence] applies the rules of equality and congruence closure, plus properties of constructors of inductive types. The [omega] tactic provides a complete decision procedure for a theory that is called quantifier-free linear arithmetic or Presburger arithmetic, depending on whom you ask. That is, [omega] proves any goal that follows from looking only at parts of that goal that can be interpreted as propositional formulas whose atomic formulas are basic comparison operations on natural numbers or integers.
adamc@132 26
adamc@132 27 The [ring] tactic solves goals by appealing to the axioms of rings or semi-rings (as in algebra), depending on the type involved. Coq developments may declare new types to be parts of rings and semi-rings by proving the associated axioms. There is a simlar tactic [field] for simplifying values in fields by conversion to fractions over rings. Both [ring] and [field] can only solve goals that are equalities. The [fourier] tactic uses Fourier's method to prove inequalities over real numbers, which are axiomatized in the Coq standard library.
adamc@132 28
adam@288 29 The %\textit{%#<i>#setoid#</i>#%}% facility makes it possible to register new equivalence relations to be understood by tactics like [rewrite]. For instance, [Prop] is registered as a setoid with the equivalence relation %``%#"#if and only if.#"#%''% The ability to register new setoids can be very useful in proofs of a kind common in math, where all reasoning is done after %``%#"#modding out by a relation.#"#%''% *)
adamc@132 30
adamc@132 31
adamc@135 32 (** * Ltac Programming Basics *)
adamc@135 33
adamc@135 34 (** We have already seen many examples of Ltac programs. In the rest of this chapter, we attempt to give a more principled introduction to the important features and design patterns.
adamc@135 35
adamc@135 36 One common use for [match] tactics is identification of subjects for case analysis, as we see in this tactic definition. *)
adamc@135 37
adamc@141 38 (* begin thide *)
adamc@135 39 Ltac find_if :=
adamc@135 40 match goal with
adamc@135 41 | [ |- if ?X then _ else _ ] => destruct X
adamc@135 42 end.
adamc@141 43 (* end thide *)
adamc@135 44
adamc@135 45 (** 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 46
adamc@135 47 Theorem hmm : forall (a b c : bool),
adamc@135 48 if a
adamc@135 49 then if b
adamc@135 50 then True
adamc@135 51 else True
adamc@135 52 else if c
adamc@135 53 then True
adamc@135 54 else True.
adamc@141 55 (* begin thide *)
adamc@135 56 intros; repeat find_if; constructor.
adamc@135 57 Qed.
adamc@141 58 (* end thide *)
adamc@135 59
adamc@135 60 (** The [repeat] that we use here is called a %\textit{%#<i>#tactical#</i>#%}%, or tactic combinator. The behavior of [repeat t] is to loop through running [t], running [t] on all generated subgoals, running [t] on %\textit{%#<i>#their#</i>#%}% generated subgoals, and so on. When [t] fails at any point in this search tree, that particular subgoal is left to be handled by later tactics. Thus, it is important never to use [repeat] with a tactic that always succeeds.
adamc@135 61
adamc@135 62 Another very useful Ltac building block is %\textit{%#<i>#context patterns#</i>#%}%. *)
adamc@135 63
adamc@141 64 (* begin thide *)
adamc@135 65 Ltac find_if_inside :=
adamc@135 66 match goal with
adamc@135 67 | [ |- context[if ?X then _ else _] ] => destruct X
adamc@135 68 end.
adamc@141 69 (* end thide *)
adamc@135 70
adamc@135 71 (** 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 72
adamc@135 73 Theorem hmm' : forall (a b c : bool),
adamc@135 74 if a
adamc@135 75 then if b
adamc@135 76 then True
adamc@135 77 else True
adamc@135 78 else if c
adamc@135 79 then True
adamc@135 80 else True.
adamc@141 81 (* begin thide *)
adamc@135 82 intros; repeat find_if_inside; constructor.
adamc@135 83 Qed.
adamc@141 84 (* end thide *)
adamc@135 85
adamc@135 86 (** We can also use [find_if_inside] to prove goals that [find_if] does not simplify sufficiently. *)
adamc@135 87
adamc@141 88 Theorem hmm2 : forall (a b : bool),
adamc@135 89 (if a then 42 else 42) = (if b then 42 else 42).
adamc@141 90 (* begin thide *)
adamc@135 91 intros; repeat find_if_inside; reflexivity.
adamc@135 92 Qed.
adamc@141 93 (* end thide *)
adamc@135 94
adam@288 95 (** 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 96
adamc@141 97 (* begin thide *)
adamc@135 98 Ltac my_tauto :=
adamc@135 99 repeat match goal with
adamc@135 100 | [ H : ?P |- ?P ] => exact H
adamc@135 101
adamc@135 102 | [ |- True ] => constructor
adamc@135 103 | [ |- _ /\ _ ] => constructor
adamc@135 104 | [ |- _ -> _ ] => intro
adamc@135 105
adamc@135 106 | [ H : False |- _ ] => destruct H
adamc@135 107 | [ H : _ /\ _ |- _ ] => destruct H
adamc@135 108 | [ H : _ \/ _ |- _ ] => destruct H
adamc@135 109
adamc@135 110 | [ H1 : ?P -> ?Q, H2 : ?P |- _ ] =>
adamc@135 111 let H := fresh "H" in
adamc@135 112 generalize (H1 H2); clear H1; intro H
adamc@135 113 end.
adamc@141 114 (* end thide *)
adamc@135 115
adamc@135 116 (** 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 [exact] tactic solves a goal completely when given a proof term of the proper type.
adamc@135 117
adam@288 118 It is also trivial to implement the %``%#"#introduction rules#"#%''% for a few of the connectives. Implementing elimination rules is only a little more work, since we must give a name for a hypothesis to [destruct].
adamc@135 119
adamc@135 120 The last rule implements modus ponens. The most interesting part is the use of the Ltac-level [let] with a [fresh] expression. [fresh] takes in a name base and returns a fresh hypothesis variable based on that name. We use the new name variable [H] as the name we assign to the result of modus ponens. The use of [generalize] changes our conclusion to be an implication from [Q]. We clear the original hypothesis and move [Q] into the context with name [H]. *)
adamc@135 121
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
adamc@135 132 (** 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
adamc@135 136 There is a related pair of two other differences that are much more important than the others. [match] has a %\textit{%#<i>#backtracking semantics for failure#</i>#%}%. In ML, pattern matching works by finding the first pattern to match and then executing its body. If the body raises an exception, then the overall match raises the same exception. In Coq, failures in case bodies instead trigger continued search through the list of cases.
adamc@135 137
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
adamc@135 151 The example shows how failure can move to a different pattern within a [match]. Failure can also trigger an attempt to find %\textit{%#<i>#a different way of matching a single pattern#</i>#%}%. Consider another example: *)
adamc@135 152
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@288 158 (** Coq prints %``%#"#[H1]#"#%''%. By applying [idtac] with an argument, a convenient debugging tool for %``%#"#leaking information out of [match]es,#"#%''% we see that this [match] first tries binding [H] to [H1], which cannot be used to prove [Q]. Nonetheless, the following variation on the tactic succeeds at proving the goal: *)
adamc@135 159
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@288 183 (** 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 [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
adamc@135 185 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 [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
adamc@135 187 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 [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
adamc@135 197 (** We see the useful [type of] operator of Ltac. This operator could not be implemented in Gallina, but it is easy to support in Ltac. We end up with [t] bound to the type of [pf]. We check that [t] is not already present. If so, we use a [generalize]/[intro] combo to add a new hypothesis proved by [pf].
adamc@135 198
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
adamc@135 206 | [ H : ?P -> ?Q, H' : ?P |- _ ] =>
adamc@135 207 generalize (H H'); clear H; intro H
adamc@135 208 | [ |- forall x, _ ] => intro
adamc@135 209
adamc@135 210 | [ H : forall x, ?P x -> _, H' : ?P ?X |- _ ] =>
adamc@135 211 extend (H X H')
adamc@135 212 end.
adamc@141 213 (* end thide *)
adamc@135 214
adamc@135 215 (** 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@135 216
adamc@135 217 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@135 218
adamc@135 219 We can check that [completer] is working properly: *)
adamc@135 220
adamc@135 221 Section firstorder.
adamc@135 222 Variable A : Set.
adamc@135 223 Variables P Q R S : A -> Prop.
adamc@135 224
adamc@135 225 Hypothesis H1 : forall x, P x -> Q x /\ R x.
adamc@135 226 Hypothesis H2 : forall x, R x -> S x.
adamc@135 227
adamc@135 228 Theorem fo : forall x, P x -> S x.
adamc@141 229 (* begin thide *)
adamc@135 230 completer.
adamc@135 231 (** [[
adamc@135 232 x : A
adamc@135 233 H : P x
adamc@135 234 H0 : Q x
adamc@135 235 H3 : R x
adamc@135 236 H4 : S x
adamc@135 237 ============================
adamc@135 238 S x
adam@302 239 ]]
adam@302 240 *)
adamc@135 241
adamc@135 242 assumption.
adamc@135 243 Qed.
adamc@141 244 (* end thide *)
adamc@135 245 End firstorder.
adamc@135 246
adamc@135 247 (** 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@135 248
adamc@141 249 (* begin thide *)
adamc@135 250 Ltac completer' :=
adamc@135 251 repeat match goal with
adamc@135 252 | [ |- _ /\ _ ] => constructor
adamc@135 253 | [ H : _ /\ _ |- _ ] => destruct H
adamc@135 254 | [ H : ?P -> _, H' : ?P |- _ ] =>
adamc@135 255 generalize (H H'); clear H; intro H
adamc@135 256 | [ |- forall x, _ ] => intro
adamc@135 257
adamc@135 258 | [ H : forall x, ?P x -> _, H' : ?P ?X |- _ ] =>
adamc@135 259 extend (H X H')
adamc@135 260 end.
adamc@141 261 (* end thide *)
adamc@135 262
adamc@135 263 (** 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@135 264
adamc@135 265 Section firstorder'.
adamc@135 266 Variable A : Set.
adamc@135 267 Variables P Q R S : A -> Prop.
adamc@135 268
adamc@135 269 Hypothesis H1 : forall x, P x -> Q x /\ R x.
adamc@135 270 Hypothesis H2 : forall x, R x -> S x.
adamc@135 271
adamc@135 272 Theorem fo' : forall x, P x -> S x.
adamc@141 273 (* begin thide *)
adamc@135 274 (** [[
adamc@135 275 completer'.
adamc@220 276
adamc@205 277 ]]
adamc@205 278
adamc@135 279 Coq loops forever at this point. What went wrong? *)
adamc@220 280
adamc@135 281 Abort.
adamc@141 282 (* end thide *)
adamc@135 283 End firstorder'.
adamc@136 284
adamc@136 285 (** A few examples should illustrate the issue. Here we see a [match]-based proof that works fine: *)
adamc@136 286
adamc@136 287 Theorem t1 : forall x : nat, x = x.
adamc@136 288 match goal with
adamc@136 289 | [ |- forall x, _ ] => trivial
adamc@136 290 end.
adamc@141 291 (* begin thide *)
adamc@136 292 Qed.
adamc@141 293 (* end thide *)
adamc@136 294
adamc@136 295 (** This one fails. *)
adamc@136 296
adamc@141 297 (* begin thide *)
adamc@136 298 Theorem t1' : forall x : nat, x = x.
adamc@136 299 (** [[
adamc@136 300 match goal with
adamc@136 301 | [ |- forall x, ?P ] => trivial
adamc@136 302 end.
adamc@136 303
adamc@136 304 User error: No matching clauses for match goal
adam@302 305 ]]
adam@302 306 *)
adamc@220 307
adamc@136 308 Abort.
adamc@141 309 (* end thide *)
adamc@136 310
adam@288 311 (** 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@136 312
adam@288 313 The Coq 8.2 release includes a special pattern form for a unification variable with an explicit set of free variables. That unification variable is then bound to a function from the free variables to the %``%#"#real#"#%''% value. In Coq 8.1 and earlier, there is no such workaround.
adamc@136 314
adam@288 315 No matter which version you use, it is important to be aware of this restriction. As we have alluded to, the restriction is the culprit behind the infinite-looping behavior of [completer']. We unintentionally match quantified facts with the modus ponens rule, circumventing the %``%#"#already present#"#%''% check and leading to different behavior, where the same fact may be added to the context repeatedly in an infinite loop. Our earlier [completer] tactic uses a modus ponens rule that matches the implication conclusion with a variable, which blocks matching against non-trivial universal quantifiers. *)
adamc@137 316
adamc@137 317
adamc@137 318 (** * Functional Programming in Ltac *)
adamc@137 319
adamc@141 320 (* EX: Write a list length function in Ltac. *)
adamc@141 321
adamc@137 322 (** 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@137 323
adamc@137 324 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@137 325
adamc@137 326 [[
adamc@137 327 Ltac length ls :=
adamc@137 328 match ls with
adamc@137 329 | nil => O
adamc@137 330 | _ :: ls' => S (length ls')
adamc@137 331 end.
adamc@137 332
adamc@137 333 Error: The reference ls' was not found in the current environment
adamc@220 334
adamc@137 335 ]]
adamc@137 336
adamc@137 337 At this point, we hopefully remember that pattern variable names must be prefixed by question marks in Ltac.
adamc@137 338
adamc@137 339 [[
adamc@137 340 Ltac length ls :=
adamc@137 341 match ls with
adamc@137 342 | nil => O
adamc@137 343 | _ :: ?ls' => S (length ls')
adamc@137 344 end.
adamc@137 345
adamc@137 346 Error: The reference S was not found in the current environment
adamc@220 347
adamc@137 348 ]]
adamc@137 349
adam@288 350 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. *)
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@301 371 We use the [pose] tactic, which extends the proof context with a new variable that is set equal to a particular term. We could also have used [idtac n] in place of [pose n], which would have printed the result without changing the context.
adamc@220 372
adamc@220 373 [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@288 410 | 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@288 414 constr:( x' :: ls'')
adamc@137 415 end in
adamc@137 416 map'.
adamc@137 417
adam@288 418 (** 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. [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@288 423 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
adamc@138 435
adamc@139 436 (** * Recursive Proof Search *)
adamc@139 437
adamc@139 438 (** 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@139 439
adam@288 440 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@139 441
adamc@141 442 (* begin thide *)
adamc@139 443 Ltac inster n :=
adamc@139 444 intuition;
adamc@139 445 match n with
adamc@139 446 | S ?n' =>
adamc@139 447 match goal with
adamc@139 448 | [ H : forall x : ?T, _, x : ?T |- _ ] => generalize (H x); inster n'
adamc@139 449 end
adamc@139 450 end.
adamc@141 451 (* end thide *)
adamc@139 452
adamc@139 453 (** [inster] begins by applying propositional simplification. Next, it checks if any chain length remains. If so, it tries all possible ways of instantiating quantified hypotheses with properly-typed local variables. It is critical to realize that, if the recursive call [inster n'] fails, then the [match goal] just seeks out another way of unifying its pattern against proof state. Thus, this small amount of code provides an elegant demonstration of how backtracking [match] enables exhaustive search.
adamc@139 454
adamc@139 455 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@139 456
adamc@139 457 Section test_inster.
adamc@139 458 Variable A : Set.
adamc@139 459 Variables P Q : A -> Prop.
adamc@139 460 Variable f : A -> A.
adamc@139 461 Variable g : A -> A -> A.
adamc@139 462
adamc@139 463 Hypothesis H1 : forall x y, P (g x y) -> Q (f x).
adamc@139 464
adamc@139 465 Theorem test_inster : forall x y, P (g x y) -> Q (f x).
adamc@220 466 inster 2.
adamc@139 467 Qed.
adamc@139 468
adamc@139 469 Hypothesis H3 : forall u v, P u /\ P v /\ u <> v -> P (g u v).
adamc@139 470 Hypothesis H4 : forall u, Q (f u) -> P u /\ P (f u).
adamc@139 471
adamc@139 472 Theorem test_inster2 : forall x y, x <> y -> P x -> Q (f y) -> Q (f x).
adamc@220 473 inster 3.
adamc@139 474 Qed.
adamc@139 475 End test_inster.
adamc@139 476
adam@288 477 (** The style employed in the definition of [inster] can seem very counterintuitive to functional programmers. Usually, functional programs accumulate state changes in explicit arguments to recursive functions. In Ltac, the state of the current subgoal is always implicit. Nonetheless, in contrast to general imperative programming, it is easy to undo any changes to this state, and indeed such %``%#"#undoing#"#%''% happens automatically at failures within [match]es. In this way, Ltac programming is similar to programming in Haskell with a stateful failure monad that supports a composition operator along the lines of the [first] tactical.
adamc@140 478
adam@288 479 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@140 480
adam@288 481 This procedure is inspired by one for separation logic, where conjuncts in formulas are thought of as %``%#"#resources,#"#%''% such that we lose no completeness by %``%#"#crossing out#"#%''% equal conjuncts on the two sides of an implication. This process is complicated by the fact that, for reasons of modularity, our formulas can have arbitrary nested tree structure (branching at conjunctions) and may include existential quantifiers. It is helpful for the matching process to %``%#"#go under#"#%''% quantifiers and in fact decide how to instantiate existential quantifiers in the conclusion.
adamc@140 482
adam@288 483 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@138 484
adamc@138 485 Definition imp (P1 P2 : Prop) := P1 -> P2.
adamc@140 486 Infix "-->" := imp (no associativity, at level 95).
adamc@140 487 Ltac imp := unfold imp; firstorder.
adamc@138 488
adamc@140 489 (** These lemmas about [imp] will be useful in the tactic that we will write. *)
adamc@138 490
adamc@138 491 Theorem and_True_prem : forall P Q,
adamc@138 492 (P /\ True --> Q)
adamc@138 493 -> (P --> Q).
adamc@138 494 imp.
adamc@138 495 Qed.
adamc@138 496
adamc@138 497 Theorem and_True_conc : forall P Q,
adamc@138 498 (P --> Q /\ True)
adamc@138 499 -> (P --> Q).
adamc@138 500 imp.
adamc@138 501 Qed.
adamc@138 502
adamc@138 503 Theorem assoc_prem1 : forall P Q R S,
adamc@138 504 (P /\ (Q /\ R) --> S)
adamc@138 505 -> ((P /\ Q) /\ R --> S).
adamc@138 506 imp.
adamc@138 507 Qed.
adamc@138 508
adamc@138 509 Theorem assoc_prem2 : forall P Q R S,
adamc@138 510 (Q /\ (P /\ R) --> S)
adamc@138 511 -> ((P /\ Q) /\ R --> S).
adamc@138 512 imp.
adamc@138 513 Qed.
adamc@138 514
adamc@138 515 Theorem comm_prem : forall P Q R,
adamc@138 516 (P /\ Q --> R)
adamc@138 517 -> (Q /\ P --> R).
adamc@138 518 imp.
adamc@138 519 Qed.
adamc@138 520
adamc@138 521 Theorem assoc_conc1 : forall P Q R S,
adamc@138 522 (S --> P /\ (Q /\ R))
adamc@138 523 -> (S --> (P /\ Q) /\ R).
adamc@138 524 imp.
adamc@138 525 Qed.
adamc@138 526
adamc@138 527 Theorem assoc_conc2 : forall P Q R S,
adamc@138 528 (S --> Q /\ (P /\ R))
adamc@138 529 -> (S --> (P /\ Q) /\ R).
adamc@138 530 imp.
adamc@138 531 Qed.
adamc@138 532
adamc@138 533 Theorem comm_conc : forall P Q R,
adamc@138 534 (R --> P /\ Q)
adamc@138 535 -> (R --> Q /\ P).
adamc@138 536 imp.
adamc@138 537 Qed.
adamc@138 538
adam@288 539 (** 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@140 540
adamc@141 541 (* begin thide *)
adamc@138 542 Ltac search_prem tac :=
adamc@138 543 let rec search P :=
adamc@138 544 tac
adamc@138 545 || (apply and_True_prem; tac)
adamc@138 546 || match P with
adamc@138 547 | ?P1 /\ ?P2 =>
adamc@138 548 (apply assoc_prem1; search P1)
adamc@138 549 || (apply assoc_prem2; search P2)
adamc@138 550 end
adamc@138 551 in match goal with
adamc@138 552 | [ |- ?P /\ _ --> _ ] => search P
adamc@138 553 | [ |- _ /\ ?P --> _ ] => apply comm_prem; search P
adamc@138 554 | [ |- _ --> _ ] => progress (tac || (apply and_True_prem; tac))
adamc@138 555 end.
adamc@138 556
adamc@140 557 (** To understand how [search_prem] works, we turn first to the final [match]. If the premise begins with a conjunction, we call the [search] procedure on each of the conjuncts, or only the first conjunct, if that already yields a case where [tac] does not fail. [search P] expects and maintains the invariant that the premise is of the form [P /\ Q] for some [Q]. We pass [P] explicitly as a kind of decreasing induction measure, to avoid looping forever when [tac] always fails. The second [match] case calls a commutativity lemma to realize this invariant, before passing control to [search]. The final [match] case tries applying [tac] directly and then, if that fails, changes the form of the goal by adding an extraneous [True] conjunct and calls [tac] again.
adamc@140 558
adamc@140 559 [search] itself tries the same tricks as in the last case of the final [match]. Additionally, if neither works, it checks if [P] is a conjunction. If so, it calls itself recursively on each conjunct, first applying associativity lemmas to maintain the goal-form invariant.
adamc@140 560
adamc@140 561 We will also want a dual function [search_conc], which does tree search through an [imp] conclusion. *)
adamc@140 562
adamc@138 563 Ltac search_conc tac :=
adamc@138 564 let rec search P :=
adamc@138 565 tac
adamc@138 566 || (apply and_True_conc; tac)
adamc@138 567 || match P with
adamc@138 568 | ?P1 /\ ?P2 =>
adamc@138 569 (apply assoc_conc1; search P1)
adamc@138 570 || (apply assoc_conc2; search P2)
adamc@138 571 end
adamc@138 572 in match goal with
adamc@138 573 | [ |- _ --> ?P /\ _ ] => search P
adamc@138 574 | [ |- _ --> _ /\ ?P ] => apply comm_conc; search P
adamc@138 575 | [ |- _ --> _ ] => progress (tac || (apply and_True_conc; tac))
adamc@138 576 end.
adamc@138 577
adamc@140 578 (** 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@140 579
adamc@138 580 Theorem False_prem : forall P Q,
adamc@138 581 False /\ P --> Q.
adamc@138 582 imp.
adamc@138 583 Qed.
adamc@138 584
adamc@138 585 Theorem True_conc : forall P Q : Prop,
adamc@138 586 (P --> Q)
adamc@138 587 -> (P --> True /\ Q).
adamc@138 588 imp.
adamc@138 589 Qed.
adamc@138 590
adamc@138 591 Theorem Match : forall P Q R : Prop,
adamc@138 592 (Q --> R)
adamc@138 593 -> (P /\ Q --> P /\ R).
adamc@138 594 imp.
adamc@138 595 Qed.
adamc@138 596
adamc@138 597 Theorem ex_prem : forall (T : Type) (P : T -> Prop) (Q R : Prop),
adamc@138 598 (forall x, P x /\ Q --> R)
adamc@138 599 -> (ex P /\ Q --> R).
adamc@138 600 imp.
adamc@138 601 Qed.
adamc@138 602
adamc@138 603 Theorem ex_conc : forall (T : Type) (P : T -> Prop) (Q R : Prop) x,
adamc@138 604 (Q --> P x /\ R)
adamc@138 605 -> (Q --> ex P /\ R).
adamc@138 606 imp.
adamc@138 607 Qed.
adamc@138 608
adam@288 609 (** We will also want a %``%#"#base case#"#%''% lemma for finishing proofs where cancelation has removed every constituent of the conclusion. *)
adamc@140 610
adamc@138 611 Theorem imp_True : forall P,
adamc@138 612 P --> True.
adamc@138 613 imp.
adamc@138 614 Qed.
adamc@138 615
adamc@220 616 (** Our final [matcher] tactic is now straightforward. First, we [intros] all variables into scope. Then we attempt simple premise simplifications, finishing the proof upon finding [False] and eliminating any existential quantifiers that we find. After that, we search through the conclusion. We remove [True] conjuncts, remove existential quantifiers by introducing unification variables for their bound variables, and search for matching premises to cancel. Finally, when no more progress is made, we see if the goal has become trivial and can be solved by [imp_True]. In each case, we use the tactic [simple apply] in place of [apply] to use a simpler, less expensive unification algorithm. *)
adamc@140 617
adamc@138 618 Ltac matcher :=
adamc@138 619 intros;
adam@288 620 repeat search_prem ltac:( simple apply False_prem || ( simple apply ex_prem; intro));
adam@288 621 repeat search_conc ltac:( simple apply True_conc || simple eapply ex_conc
adam@288 622 || search_prem ltac:( simple apply Match));
adamc@204 623 try simple apply imp_True.
adamc@141 624 (* end thide *)
adamc@140 625
adamc@140 626 (** Our tactic succeeds at proving a simple example. *)
adamc@138 627
adamc@138 628 Theorem t2 : forall P Q : Prop,
adamc@138 629 Q /\ (P /\ False) /\ P --> P /\ Q.
adamc@138 630 matcher.
adamc@138 631 Qed.
adamc@138 632
adamc@140 633 (** In the generated proof, we find a trace of the workings of the search tactics. *)
adamc@140 634
adamc@140 635 Print t2.
adamc@220 636 (** %\vspace{-.15in}% [[
adamc@140 637 t2 =
adamc@140 638 fun P Q : Prop =>
adamc@140 639 comm_prem (assoc_prem1 (assoc_prem2 (False_prem (P:=P /\ P /\ Q) (P /\ Q))))
adamc@140 640 : forall P Q : Prop, Q /\ (P /\ False) /\ P --> P /\ Q
adamc@220 641
adamc@220 642 ]]
adamc@140 643
adamc@220 644 We can also see that [matcher] is well-suited for cases where some human intervention is needed after the automation finishes. *)
adamc@140 645
adamc@138 646 Theorem t3 : forall P Q R : Prop,
adamc@138 647 P /\ Q --> Q /\ R /\ P.
adamc@138 648 matcher.
adamc@140 649 (** [[
adamc@140 650 ============================
adamc@140 651 True --> R
adamc@220 652
adamc@140 653 ]]
adamc@140 654
adamc@140 655 [matcher] canceled those conjuncts that it was able to cancel, leaving a simplified subgoal for us, much as [intuition] does. *)
adamc@220 656
adamc@138 657 Abort.
adamc@138 658
adamc@140 659 (** [matcher] even succeeds at guessing quantifier instantiations. It is the unification that occurs in uses of the [Match] lemma that does the real work here. *)
adamc@140 660
adamc@138 661 Theorem t4 : forall (P : nat -> Prop) Q, (exists x, P x /\ Q) --> Q /\ (exists x, P x).
adamc@138 662 matcher.
adamc@138 663 Qed.
adamc@138 664
adamc@140 665 Print t4.
adamc@220 666 (** %\vspace{-.15in}% [[
adamc@140 667 t4 =
adamc@140 668 fun (P : nat -> Prop) (Q : Prop) =>
adamc@140 669 and_True_prem
adamc@140 670 (ex_prem (P:=fun x : nat => P x /\ Q)
adamc@140 671 (fun x : nat =>
adamc@140 672 assoc_prem2
adamc@140 673 (Match (P:=Q)
adamc@140 674 (and_True_conc
adamc@140 675 (ex_conc (fun x0 : nat => P x0) x
adamc@140 676 (Match (P:=P x) (imp_True (P:=True))))))))
adamc@140 677 : forall (P : nat -> Prop) (Q : Prop),
adamc@140 678 (exists x : nat, P x /\ Q) --> Q /\ (exists x : nat, P x)
adam@302 679 ]]
adam@302 680 *)
adamc@234 681
adamc@234 682
adamc@234 683 (** * Creating Unification Variables *)
adamc@234 684
adamc@234 685 (** A final useful ingredient in tactic crafting is the ability to allocate new unification variables explicitly. Tactics like [eauto] introduce unification variable internally to support flexible proof search. While [eauto] and its relatives do %\textit{%#<i>#backward#</i>#%}% reasoning, we often want to do similar %\textit{%#<i>#forward#</i>#%}% reasoning, where unification variables can be useful for similar reasons.
adamc@234 686
adamc@234 687 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@234 688
adamc@234 689 Before we are ready to write a tactic, we can try out its ingredients one at a time. *)
adamc@234 690
adamc@234 691 Theorem t5 : (forall x : nat, S x > x) -> 2 > 1.
adamc@234 692 intros.
adamc@234 693
adamc@234 694 (** [[
adamc@234 695 H : forall x : nat, S x > x
adamc@234 696 ============================
adamc@234 697 2 > 1
adamc@234 698
adamc@234 699 ]]
adamc@234 700
adamc@234 701 To instantiate [H] generically, we first need to name the value to be used for [x]. *)
adamc@234 702
adamc@234 703 evar (y : nat).
adamc@234 704
adamc@234 705 (** [[
adamc@234 706 H : forall x : nat, S x > x
adamc@234 707 y := ?279 : nat
adamc@234 708 ============================
adamc@234 709 2 > 1
adamc@234 710
adamc@234 711 ]]
adamc@234 712
adamc@234 713 The proof context is extended with a new variable [y], which has been assigned to be equal to a fresh unification variable [?279]. We want to instantiate [H] with [?279]. To get ahold of the new unification variable, rather than just its alias [y], we perform a trivial call-by-value reduction in the expression [y]. In particular, we only request the use of one reduction rule, [delta], which deals with definition unfolding. We pass a flag further stipulating that only the definition of [y] be unfolded. This is a simple trick for getting at the value of a synonym variable. *)
adamc@234 714
adamc@234 715 let y' := eval cbv delta [y] in y in
adamc@234 716 clear y; generalize (H y').
adamc@234 717
adamc@234 718 (** [[
adamc@234 719 H : forall x : nat, S x > x
adamc@234 720 ============================
adamc@234 721 S ?279 > ?279 -> 2 > 1
adamc@234 722
adamc@234 723 ]]
adamc@234 724
adamc@234 725 Our instantiation was successful. We can finish by using the refined formula to replace the original. *)
adamc@234 726
adamc@234 727 clear H; intro H.
adamc@234 728
adamc@234 729 (** [[
adamc@234 730 H : S ?281 > ?281
adamc@234 731 ============================
adamc@234 732 2 > 1
adamc@234 733
adamc@234 734 ]]
adamc@234 735
adamc@234 736 We can finish the proof by using [apply]'s unification to figure out the proper value of [?281]. (The original unification variable was replaced by another, as often happens in the internals of the various tactics' implementations.) *)
adamc@234 737
adamc@234 738 apply H.
adamc@234 739 Qed.
adamc@234 740
adamc@234 741 (** Now we can write a tactic that encapsulates the pattern we just employed, instantiating all quantifiers of a particular hypothesis. *)
adamc@234 742
adamc@234 743 Ltac insterU H :=
adamc@234 744 repeat match type of H with
adamc@234 745 | forall x : ?T, _ =>
adamc@234 746 let x := fresh "x" in
adamc@234 747 evar (x : T);
adamc@234 748 let x' := eval cbv delta [x] in x in
adamc@234 749 clear x; generalize (H x'); clear H; intro H
adamc@234 750 end.
adamc@234 751
adamc@234 752 Theorem t5' : (forall x : nat, S x > x) -> 2 > 1.
adamc@234 753 intro H; insterU H; apply H.
adamc@234 754 Qed.
adamc@234 755
adamc@234 756 (** This particular example is somewhat silly, since [apply] by itself would have solved the goal originally. Separate forward reasoning is more useful on hypotheses that end in existential quantifications. Before we go through an example, it is useful to define a variant of [insterU] that does not clear the base hypothesis we pass to it. *)
adamc@234 757
adamc@234 758 Ltac insterKeep H :=
adamc@234 759 let H' := fresh "H'" in
adamc@234 760 generalize H; intro H'; insterU H'.
adamc@234 761
adamc@234 762 Section t6.
adamc@234 763 Variables A B : Type.
adamc@234 764 Variable P : A -> B -> Prop.
adamc@234 765 Variable f : A -> A -> A.
adamc@234 766 Variable g : B -> B -> B.
adamc@234 767
adamc@234 768 Hypothesis H1 : forall v, exists u, P v u.
adamc@234 769 Hypothesis H2 : forall v1 u1 v2 u2,
adamc@234 770 P v1 u1
adamc@234 771 -> P v2 u2
adamc@234 772 -> P (f v1 v2) (g u1 u2).
adamc@234 773
adamc@234 774 Theorem t6 : forall v1 v2, exists u1, exists u2, P (f v1 v2) (g u1 u2).
adamc@234 775 intros.
adamc@234 776
adamc@234 777 (** Neither [eauto] nor [firstorder] is clever enough to prove this goal. We can help out by doing some of the work with quantifiers ourselves. *)
adamc@234 778
adamc@234 779 do 2 insterKeep H1.
adamc@234 780
adamc@234 781 (** Our proof state is extended with two generic instances of [H1].
adamc@234 782
adamc@234 783 [[
adamc@234 784 H' : exists u : B, P ?4289 u
adamc@234 785 H'0 : exists u : B, P ?4288 u
adamc@234 786 ============================
adamc@234 787 exists u1 : B, exists u2 : B, P (f v1 v2) (g u1 u2)
adamc@234 788
adamc@234 789 ]]
adamc@234 790
adamc@234 791 [eauto] still cannot prove the goal, so we eliminate the two new existential quantifiers. *)
adamc@234 792
adamc@234 793 repeat match goal with
adamc@234 794 | [ H : ex _ |- _ ] => destruct H
adamc@234 795 end.
adamc@234 796
adamc@234 797 (** Now the goal is simple enough to solve by logic programming. *)
adamc@234 798
adamc@234 799 eauto.
adamc@234 800 Qed.
adamc@234 801 End t6.
adamc@234 802
adamc@234 803 (** 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@234 804
adamc@234 805 Section t7.
adamc@234 806 Variables A B : Type.
adamc@234 807 Variable Q : A -> Prop.
adamc@234 808 Variable P : A -> B -> Prop.
adamc@234 809 Variable f : A -> A -> A.
adamc@234 810 Variable g : B -> B -> B.
adamc@234 811
adamc@234 812 Hypothesis H1 : forall v, Q v -> exists u, P v u.
adamc@234 813 Hypothesis H2 : forall v1 u1 v2 u2,
adamc@234 814 P v1 u1
adamc@234 815 -> P v2 u2
adamc@234 816 -> P (f v1 v2) (g u1 u2).
adamc@234 817
adam@297 818 Theorem t7 : forall v1 v2, Q v1 -> Q v2 -> exists u1, exists u2, P (f v1 v2) (g u1 u2).
adamc@234 819 intros; do 2 insterKeep H1;
adamc@234 820 repeat match goal with
adamc@234 821 | [ H : ex _ |- _ ] => destruct H
adamc@234 822 end; eauto.
adamc@234 823
adamc@234 824 (** This proof script does not hit any errors until the very end, when an error message like this one is displayed.
adamc@234 825
adamc@234 826 [[
adamc@234 827 No more subgoals but non-instantiated existential variables :
adamc@234 828 Existential 1 =
adamc@234 829 ?4384 : [A : Type
adamc@234 830 B : Type
adamc@234 831 Q : A -> Prop
adamc@234 832 P : A -> B -> Prop
adamc@234 833 f : A -> A -> A
adamc@234 834 g : B -> B -> B
adamc@234 835 H1 : forall v : A, Q v -> exists u : B, P v u
adamc@234 836 H2 : forall (v1 : A) (u1 : B) (v2 : A) (u2 : B),
adamc@234 837 P v1 u1 -> P v2 u2 -> P (f v1 v2) (g u1 u2)
adamc@234 838 v1 : A
adamc@234 839 v2 : A
adamc@234 840 H : Q v1
adamc@234 841 H0 : Q v2
adamc@234 842 H' : Q v2 -> exists u : B, P v2 u |- Q v2]
adamc@234 843
adamc@234 844 ]]
adamc@234 845
adam@288 846 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@234 847
adamc@234 848 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@234 849
adamc@234 850 Abort.
adamc@234 851 End t7.
adamc@234 852
adamc@234 853 Reset insterU.
adamc@234 854
adamc@234 855 (** We can redefine [insterU] to treat implications differently. In particular, we pattern-match on the type of the type [T] in [forall x : ?T, ...]. If [T] has type [Prop], then [x]'s instantiation should be thought of as a proof. Thus, instead of picking a new unification variable for it, we instead apply a user-supplied tactic [tac]. It is important that we end this special [Prop] case with [|| fail 1], so that, if [tac] fails to prove [T], we abort the instantiation, rather than continuing on to the default quantifier handling. *)
adamc@234 856
adamc@234 857 Ltac insterU tac H :=
adamc@234 858 repeat match type of H with
adamc@234 859 | forall x : ?T, _ =>
adamc@234 860 match type of T with
adamc@234 861 | Prop =>
adamc@234 862 (let H' := fresh "H'" in
adamc@234 863 assert (H' : T); [
adamc@234 864 solve [ tac ]
adamc@234 865 | generalize (H H'); clear H H'; intro H ])
adamc@234 866 || fail 1
adamc@234 867 | _ =>
adamc@234 868 let x := fresh "x" in
adamc@234 869 evar (x : T);
adamc@234 870 let x' := eval cbv delta [x] in x in
adamc@234 871 clear x; generalize (H x'); clear H; intro H
adamc@234 872 end
adamc@234 873 end.
adamc@234 874
adamc@234 875 Ltac insterKeep tac H :=
adamc@234 876 let H' := fresh "H'" in
adamc@234 877 generalize H; intro H'; insterU tac H'.
adamc@234 878
adamc@234 879 Section t7.
adamc@234 880 Variables A B : Type.
adamc@234 881 Variable Q : A -> Prop.
adamc@234 882 Variable P : A -> B -> Prop.
adamc@234 883 Variable f : A -> A -> A.
adamc@234 884 Variable g : B -> B -> B.
adamc@234 885
adamc@234 886 Hypothesis H1 : forall v, Q v -> exists u, P v u.
adamc@234 887 Hypothesis H2 : forall v1 u1 v2 u2,
adamc@234 888 P v1 u1
adamc@234 889 -> P v2 u2
adamc@234 890 -> P (f v1 v2) (g u1 u2).
adamc@234 891
adamc@234 892 Theorem t6 : forall v1 v2, Q v1 -> Q v2 -> exists u1, exists u2, P (f v1 v2) (g u1 u2).
adamc@234 893
adamc@234 894 (** 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@234 895
adamc@234 896 intros; do 2 insterKeep ltac:(idtac; match goal with
adamc@234 897 | [ H : Q ?v |- _ ] =>
adamc@234 898 match goal with
adamc@234 899 | [ _ : context[P v _] |- _ ] => fail 1
adamc@234 900 | _ => apply H
adamc@234 901 end
adamc@234 902 end) H1;
adamc@234 903 repeat match goal with
adamc@234 904 | [ H : ex _ |- _ ] => destruct H
adamc@234 905 end; eauto.
adamc@234 906 Qed.
adamc@234 907 End t7.
adamc@234 908
adamc@234 909 (** It is often useful to instantiate existential variables explicitly. A built-in tactic provides one way of doing so. *)
adamc@234 910
adamc@234 911 Theorem t8 : exists p : nat * nat, fst p = 3.
adamc@234 912 econstructor; instantiate (1 := (3, 2)); reflexivity.
adamc@234 913 Qed.
adamc@234 914
adamc@234 915 (** 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@234 916
adamc@234 917 The [instantiate] tactic can be convenient for exploratory proving, but it leads to very brittle proof scripts that are unlikely to adapt to changing theorem statements. It is often more helpful to have a tactic that can be used to assign a value to a term that is known to be an existential. By employing a roundabout implementation technique, we can build a tactic that generalizes this functionality. In particular, our tactic [equate] will assert that two terms are equal. If one of the terms happens to be an existential, then it will be replaced everywhere with the other term. *)
adamc@234 918
adamc@234 919 Ltac equate x y :=
adamc@234 920 let H := fresh "H" in
adamc@234 921 assert (H : x = y); [ reflexivity | clear H ].
adamc@234 922
adamc@234 923 (** [equate] fails if it is not possible to prove [x = y] by [reflexivity]. We perform the proof only for its unification side effects, clearing the fact [x = y] afterward. With [equate], we can build a less brittle version of the prior example. *)
adamc@234 924
adamc@234 925 Theorem t9 : exists p : nat * nat, fst p = 3.
adamc@234 926 econstructor; match goal with
adamc@234 927 | [ |- fst ?x = 3 ] => equate x (3, 2)
adamc@234 928 end; reflexivity.
adamc@234 929 Qed.