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Strengthen last theorem of Coinductive
author Adam Chlipala <adam@chlipala.net>
date Sun, 08 Apr 2012 11:11:27 -0400
parents 549d604c3d16
children b911d0df5eee
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
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@328 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.
adamc@132 26
adam@328 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 %\index{tactics!field}\coqdockw{%#<tt>#field#</tt>#%}% for simplifying values in fields by conversion to fractions over rings. Both [ring] and %\coqdockw{%#<tt>#field#</tt>#%}% can only solve goals that are equalities. The %\index{tactics!fourier}\coqdockw{%#<tt>#fourier#</tt>#%}% tactic uses Fourier's method to prove inequalities over real numbers, which are axiomatized in the Coq standard library.
adamc@132 28
adam@328 29 The %\index{setoids}\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.#"#%''%
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@328 62 (** The %\index{tactics!repeat}%[repeat] that we use here is called a %\index{tactical}\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 63
adam@328 64 Another very useful Ltac building block is %\index{context patterns}\textit{%#<i>#context patterns#</i>#%}%. *)
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@328 120 The last rule implements modus ponens, using a tactic %\index{tactics!specialize}\coqdockw{%#<tt>#specialize#</tt>#%}% 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@328 136 There is a related pair of two other differences that are much more important than the others. The [match] construct 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@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@328 197 (** 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 a proof given as argument).
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@328 259 (** 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@328 309 (** 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@288 311 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.
adamc@136 312
adam@288 313 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@137 314
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@328 350 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
adam@328 377 (* begin hide *)
adamc@137 378 Reset length.
adam@328 379 (* end hide *)
adam@328 380 (** %\noindent\coqdockw{%#<tt>#Reset#</tt>#%}% [length.] *)
adamc@137 381
adamc@137 382 (** 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@137 383
adamc@137 384 Ltac length ls :=
adamc@137 385 match ls with
adamc@137 386 | nil => O
adamc@137 387 | _ :: ?ls' =>
adamc@137 388 let ls'' := length ls' in
adamc@137 389 constr:(S ls'')
adamc@137 390 end.
adamc@137 391
adamc@137 392 Goal False.
adamc@137 393 let n := length (1 :: 2 :: 3 :: nil) in
adamc@137 394 pose n.
adamc@137 395 (** [[
adamc@137 396 n := 3 : nat
adamc@137 397 ============================
adamc@137 398 False
adam@302 399 ]]
adam@302 400 *)
adamc@220 401
adamc@137 402 Abort.
adamc@141 403 (* end thide *)
adamc@141 404
adamc@141 405 (* EX: Write a list map function in Ltac. *)
adamc@137 406
adamc@137 407 (** We can also use anonymous function expressions and local function definitions in Ltac, as this example of a standard list [map] function shows. *)
adamc@137 408
adamc@141 409 (* begin thide *)
adamc@137 410 Ltac map T f :=
adamc@137 411 let rec map' ls :=
adamc@137 412 match ls with
adam@288 413 | nil => constr:( @nil T)
adamc@137 414 | ?x :: ?ls' =>
adamc@137 415 let x' := f x in
adamc@137 416 let ls'' := map' ls' in
adam@288 417 constr:( x' :: ls'')
adamc@137 418 end in
adamc@137 419 map'.
adamc@137 420
adam@328 421 (** 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@137 422
adam@288 423 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@137 424
adamc@137 425 Goal False.
adam@288 426 let ls := map (nat * nat)%type ltac:(fun x => constr:( x, x)) (1 :: 2 :: 3 :: nil) in
adamc@137 427 pose ls.
adamc@137 428 (** [[
adamc@137 429 l := (1, 1) :: (2, 2) :: (3, 3) :: nil : list (nat * nat)
adamc@137 430 ============================
adamc@137 431 False
adam@302 432 ]]
adam@302 433 *)
adamc@220 434
adamc@137 435 Abort.
adamc@141 436 (* end thide *)
adamc@137 437
adam@328 438 (** One other gotcha shows up when we want to debug our Ltac functional programs. We might expect the following code to work, to give us a version of [length] that prints a debug trace of the arguments it is called with. *)
adam@328 439
adam@334 440 (* begin thide *)
adam@328 441 (* begin hide *)
adam@328 442 Reset length.
adam@328 443 (* end hide *)
adam@328 444 (** %\noindent\coqdockw{%#<tt>#Reset#</tt>#%}% [length.] *)
adam@328 445
adam@328 446 Ltac length ls :=
adam@328 447 idtac ls;
adam@328 448 match ls with
adam@328 449 | nil => O
adam@328 450 | _ :: ?ls' =>
adam@328 451 let ls'' := length ls' in
adam@328 452 constr:(S ls'')
adam@328 453 end.
adam@328 454
adam@328 455 (** Coq accepts the tactic definition, but the code is fatally flawed and will always lead to dynamic type errors. *)
adam@328 456
adam@328 457 Goal False.
adam@328 458 (** %\vspace{-.15in}%[[
adam@328 459 let n := length (1 :: 2 :: 3 :: nil) in
adam@328 460 pose n.
adam@328 461 ]]
adam@328 462
adam@328 463 <<
adam@328 464 Error: variable n should be bound to a term.
adam@328 465 >> *)
adam@328 466 Abort.
adam@328 467
adam@356 468 (** 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 %\emph{%#<i>#the code of programs in an imperative language#</i>#%}%, 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@328 469
adam@328 470 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@328 471
adam@328 472 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}\emph{%#<i>#continuation-passing style#</i>#%}%, a program structuring idea popular in functional programming. *)
adam@328 473
adam@328 474 (* begin hide *)
adam@328 475 Reset length.
adam@328 476 (* end hide *)
adam@328 477 (** %\noindent\coqdockw{%#<tt>#Reset#</tt>#%}% [length.] *)
adam@328 478
adam@328 479 Ltac length ls k :=
adam@328 480 idtac ls;
adam@328 481 match ls with
adam@328 482 | nil => k O
adam@328 483 | _ :: ?ls' => length ls' ltac:(fun n => k (S n))
adam@328 484 end.
adam@334 485 (* end thide *)
adam@328 486
adam@328 487 (** The new [length] takes a new input: a %\emph{%#<i>#continuation#</i>#%}% [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@328 488
adam@334 489 (* begin thide *)
adam@328 490 Goal False.
adam@328 491 length (1 :: 2 :: 3 :: nil) ltac:(fun n => pose n).
adam@328 492 (** [[
adam@328 493 (1 :: 2 :: 3 :: nil)
adam@328 494 (2 :: 3 :: nil)
adam@328 495 (3 :: nil)
adam@328 496 nil
adam@328 497 ]]
adam@328 498 *)
adam@328 499 Abort.
adam@334 500 (* end thide *)
adam@328 501
adam@328 502 (** 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@328 503
adamc@138 504
adamc@139 505 (** * Recursive Proof Search *)
adamc@139 506
adamc@139 507 (** 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@139 508
adam@288 509 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@139 510
adamc@141 511 (* begin thide *)
adamc@139 512 Ltac inster n :=
adamc@139 513 intuition;
adamc@139 514 match n with
adamc@139 515 | S ?n' =>
adamc@139 516 match goal with
adamc@139 517 | [ H : forall x : ?T, _, x : ?T |- _ ] => generalize (H x); inster n'
adamc@139 518 end
adamc@139 519 end.
adamc@141 520 (* end thide *)
adamc@139 521
adam@328 522 (** The tactic 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@139 523
adamc@139 524 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@139 525
adamc@139 526 Section test_inster.
adamc@139 527 Variable A : Set.
adamc@139 528 Variables P Q : A -> Prop.
adamc@139 529 Variable f : A -> A.
adamc@139 530 Variable g : A -> A -> A.
adamc@139 531
adamc@139 532 Hypothesis H1 : forall x y, P (g x y) -> Q (f x).
adamc@139 533
adam@328 534 Theorem test_inster : forall x, P (g x x) -> Q (f x).
adamc@220 535 inster 2.
adamc@139 536 Qed.
adamc@139 537
adamc@139 538 Hypothesis H3 : forall u v, P u /\ P v /\ u <> v -> P (g u v).
adamc@139 539 Hypothesis H4 : forall u, Q (f u) -> P u /\ P (f u).
adamc@139 540
adamc@139 541 Theorem test_inster2 : forall x y, x <> y -> P x -> Q (f y) -> Q (f x).
adamc@220 542 inster 3.
adamc@139 543 Qed.
adamc@139 544 End test_inster.
adamc@139 545
adam@328 546 (** 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. The key pieces of state include not only the form of the goal, but also decisions about the values of unification variables. These decisions are rolled back with all the other state after failure.
adamc@140 547
adam@288 548 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@140 549
adam@328 550 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@140 551
adam@288 552 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@138 553
adamc@138 554 Definition imp (P1 P2 : Prop) := P1 -> P2.
adamc@140 555 Infix "-->" := imp (no associativity, at level 95).
adamc@140 556 Ltac imp := unfold imp; firstorder.
adamc@138 557
adamc@140 558 (** These lemmas about [imp] will be useful in the tactic that we will write. *)
adamc@138 559
adamc@138 560 Theorem and_True_prem : forall P Q,
adamc@138 561 (P /\ True --> Q)
adamc@138 562 -> (P --> Q).
adamc@138 563 imp.
adamc@138 564 Qed.
adamc@138 565
adamc@138 566 Theorem and_True_conc : forall P Q,
adamc@138 567 (P --> Q /\ True)
adamc@138 568 -> (P --> Q).
adamc@138 569 imp.
adamc@138 570 Qed.
adamc@138 571
adamc@138 572 Theorem assoc_prem1 : forall P Q R S,
adamc@138 573 (P /\ (Q /\ R) --> S)
adamc@138 574 -> ((P /\ Q) /\ R --> S).
adamc@138 575 imp.
adamc@138 576 Qed.
adamc@138 577
adamc@138 578 Theorem assoc_prem2 : forall P Q R S,
adamc@138 579 (Q /\ (P /\ R) --> S)
adamc@138 580 -> ((P /\ Q) /\ R --> S).
adamc@138 581 imp.
adamc@138 582 Qed.
adamc@138 583
adamc@138 584 Theorem comm_prem : forall P Q R,
adamc@138 585 (P /\ Q --> R)
adamc@138 586 -> (Q /\ P --> R).
adamc@138 587 imp.
adamc@138 588 Qed.
adamc@138 589
adamc@138 590 Theorem assoc_conc1 : forall P Q R S,
adamc@138 591 (S --> P /\ (Q /\ R))
adamc@138 592 -> (S --> (P /\ Q) /\ R).
adamc@138 593 imp.
adamc@138 594 Qed.
adamc@138 595
adamc@138 596 Theorem assoc_conc2 : forall P Q R S,
adamc@138 597 (S --> Q /\ (P /\ R))
adamc@138 598 -> (S --> (P /\ Q) /\ R).
adamc@138 599 imp.
adamc@138 600 Qed.
adamc@138 601
adamc@138 602 Theorem comm_conc : forall P Q R,
adamc@138 603 (R --> P /\ Q)
adamc@138 604 -> (R --> Q /\ P).
adamc@138 605 imp.
adamc@138 606 Qed.
adamc@138 607
adam@288 608 (** 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@140 609
adamc@138 610 Ltac search_prem tac :=
adamc@138 611 let rec search P :=
adamc@138 612 tac
adamc@138 613 || (apply and_True_prem; tac)
adamc@138 614 || match P with
adamc@138 615 | ?P1 /\ ?P2 =>
adamc@138 616 (apply assoc_prem1; search P1)
adamc@138 617 || (apply assoc_prem2; search P2)
adamc@138 618 end
adamc@138 619 in match goal with
adamc@138 620 | [ |- ?P /\ _ --> _ ] => search P
adamc@138 621 | [ |- _ /\ ?P --> _ ] => apply comm_prem; search P
adamc@138 622 | [ |- _ --> _ ] => progress (tac || (apply and_True_prem; tac))
adamc@138 623 end.
adamc@138 624
adam@328 625 (** 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@140 626
adam@328 627 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@140 628
adamc@140 629 We will also want a dual function [search_conc], which does tree search through an [imp] conclusion. *)
adamc@140 630
adamc@138 631 Ltac search_conc tac :=
adamc@138 632 let rec search P :=
adamc@138 633 tac
adamc@138 634 || (apply and_True_conc; tac)
adamc@138 635 || match P with
adamc@138 636 | ?P1 /\ ?P2 =>
adamc@138 637 (apply assoc_conc1; search P1)
adamc@138 638 || (apply assoc_conc2; search P2)
adamc@138 639 end
adamc@138 640 in match goal with
adamc@138 641 | [ |- _ --> ?P /\ _ ] => search P
adamc@138 642 | [ |- _ --> _ /\ ?P ] => apply comm_conc; search P
adamc@138 643 | [ |- _ --> _ ] => progress (tac || (apply and_True_conc; tac))
adamc@138 644 end.
adamc@138 645
adamc@140 646 (** 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@140 647
adam@328 648 (* begin thide *)
adamc@138 649 Theorem False_prem : forall P Q,
adamc@138 650 False /\ P --> Q.
adamc@138 651 imp.
adamc@138 652 Qed.
adamc@138 653
adamc@138 654 Theorem True_conc : forall P Q : Prop,
adamc@138 655 (P --> Q)
adamc@138 656 -> (P --> True /\ Q).
adamc@138 657 imp.
adamc@138 658 Qed.
adamc@138 659
adamc@138 660 Theorem Match : forall P Q R : Prop,
adamc@138 661 (Q --> R)
adamc@138 662 -> (P /\ Q --> P /\ R).
adamc@138 663 imp.
adamc@138 664 Qed.
adamc@138 665
adamc@138 666 Theorem ex_prem : forall (T : Type) (P : T -> Prop) (Q R : Prop),
adamc@138 667 (forall x, P x /\ Q --> R)
adamc@138 668 -> (ex P /\ Q --> R).
adamc@138 669 imp.
adamc@138 670 Qed.
adamc@138 671
adamc@138 672 Theorem ex_conc : forall (T : Type) (P : T -> Prop) (Q R : Prop) x,
adamc@138 673 (Q --> P x /\ R)
adamc@138 674 -> (Q --> ex P /\ R).
adamc@138 675 imp.
adamc@138 676 Qed.
adamc@138 677
adam@288 678 (** We will also want a %``%#"#base case#"#%''% lemma for finishing proofs where cancelation has removed every constituent of the conclusion. *)
adamc@140 679
adamc@138 680 Theorem imp_True : forall P,
adamc@138 681 P --> True.
adamc@138 682 imp.
adamc@138 683 Qed.
adamc@138 684
adamc@220 685 (** 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@140 686
adamc@138 687 Ltac matcher :=
adamc@138 688 intros;
adam@288 689 repeat search_prem ltac:( simple apply False_prem || ( simple apply ex_prem; intro));
adam@288 690 repeat search_conc ltac:( simple apply True_conc || simple eapply ex_conc
adam@288 691 || search_prem ltac:( simple apply Match));
adamc@204 692 try simple apply imp_True.
adamc@141 693 (* end thide *)
adamc@140 694
adamc@140 695 (** Our tactic succeeds at proving a simple example. *)
adamc@138 696
adamc@138 697 Theorem t2 : forall P Q : Prop,
adamc@138 698 Q /\ (P /\ False) /\ P --> P /\ Q.
adamc@138 699 matcher.
adamc@138 700 Qed.
adamc@138 701
adamc@140 702 (** In the generated proof, we find a trace of the workings of the search tactics. *)
adamc@140 703
adamc@140 704 Print t2.
adamc@220 705 (** %\vspace{-.15in}% [[
adamc@140 706 t2 =
adamc@140 707 fun P Q : Prop =>
adamc@140 708 comm_prem (assoc_prem1 (assoc_prem2 (False_prem (P:=P /\ P /\ Q) (P /\ Q))))
adamc@140 709 : forall P Q : Prop, Q /\ (P /\ False) /\ P --> P /\ Q
adamc@220 710
adamc@220 711 ]]
adamc@140 712
adamc@220 713 We can also see that [matcher] is well-suited for cases where some human intervention is needed after the automation finishes. *)
adamc@140 714
adamc@138 715 Theorem t3 : forall P Q R : Prop,
adamc@138 716 P /\ Q --> Q /\ R /\ P.
adamc@138 717 matcher.
adamc@140 718 (** [[
adamc@140 719 ============================
adamc@140 720 True --> R
adamc@220 721
adamc@140 722 ]]
adamc@140 723
adam@328 724 Our tactic canceled those conjuncts that it was able to cancel, leaving a simplified subgoal for us, much as [intuition] does. *)
adamc@220 725
adamc@138 726 Abort.
adamc@138 727
adam@328 728 (** 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@140 729
adamc@138 730 Theorem t4 : forall (P : nat -> Prop) Q, (exists x, P x /\ Q) --> Q /\ (exists x, P x).
adamc@138 731 matcher.
adamc@138 732 Qed.
adamc@138 733
adamc@140 734 Print t4.
adamc@220 735 (** %\vspace{-.15in}% [[
adamc@140 736 t4 =
adamc@140 737 fun (P : nat -> Prop) (Q : Prop) =>
adamc@140 738 and_True_prem
adamc@140 739 (ex_prem (P:=fun x : nat => P x /\ Q)
adamc@140 740 (fun x : nat =>
adamc@140 741 assoc_prem2
adamc@140 742 (Match (P:=Q)
adamc@140 743 (and_True_conc
adamc@140 744 (ex_conc (fun x0 : nat => P x0) x
adamc@140 745 (Match (P:=P x) (imp_True (P:=True))))))))
adamc@140 746 : forall (P : nat -> Prop) (Q : Prop),
adamc@140 747 (exists x : nat, P x /\ Q) --> Q /\ (exists x : nat, P x)
adam@302 748 ]]
adam@302 749 *)
adamc@234 750
adamc@234 751
adamc@234 752 (** * Creating Unification Variables *)
adamc@234 753
adamc@234 754 (** 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@234 755
adam@328 756 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@234 757
adamc@234 758 Before we are ready to write a tactic, we can try out its ingredients one at a time. *)
adamc@234 759
adamc@234 760 Theorem t5 : (forall x : nat, S x > x) -> 2 > 1.
adamc@234 761 intros.
adamc@234 762
adamc@234 763 (** [[
adamc@234 764 H : forall x : nat, S x > x
adamc@234 765 ============================
adamc@234 766 2 > 1
adamc@234 767
adamc@234 768 ]]
adamc@234 769
adam@328 770 To instantiate [H] generically, we first need to name the value to be used for [x].%\index{tactics!evar}% *)
adamc@234 771
adamc@234 772 evar (y : nat).
adamc@234 773
adamc@234 774 (** [[
adamc@234 775 H : forall x : nat, S x > x
adamc@234 776 y := ?279 : nat
adamc@234 777 ============================
adamc@234 778 2 > 1
adamc@234 779
adamc@234 780 ]]
adamc@234 781
adam@328 782 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@234 783
adam@328 784 let y' := eval unfold y in y in
adamc@234 785 clear y; generalize (H y').
adamc@234 786
adamc@234 787 (** [[
adamc@234 788 H : forall x : nat, S x > x
adamc@234 789 ============================
adamc@234 790 S ?279 > ?279 -> 2 > 1
adamc@234 791
adamc@234 792 ]]
adamc@234 793
adamc@234 794 Our instantiation was successful. We can finish by using the refined formula to replace the original. *)
adamc@234 795
adamc@234 796 clear H; intro H.
adamc@234 797
adamc@234 798 (** [[
adamc@234 799 H : S ?281 > ?281
adamc@234 800 ============================
adamc@234 801 2 > 1
adamc@234 802
adamc@234 803 ]]
adamc@234 804
adamc@234 805 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@234 806
adamc@234 807 apply H.
adamc@234 808 Qed.
adamc@234 809
adamc@234 810 (** Now we can write a tactic that encapsulates the pattern we just employed, instantiating all quantifiers of a particular hypothesis. *)
adamc@234 811
adamc@234 812 Ltac insterU H :=
adamc@234 813 repeat match type of H with
adamc@234 814 | forall x : ?T, _ =>
adamc@234 815 let x := fresh "x" in
adamc@234 816 evar (x : T);
adam@328 817 let x' := eval unfold x in x in
adam@328 818 clear x; specialize (H x')
adamc@234 819 end.
adamc@234 820
adamc@234 821 Theorem t5' : (forall x : nat, S x > x) -> 2 > 1.
adamc@234 822 intro H; insterU H; apply H.
adamc@234 823 Qed.
adamc@234 824
adam@328 825 (** 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@234 826
adamc@234 827 Ltac insterKeep H :=
adamc@234 828 let H' := fresh "H'" in
adamc@234 829 generalize H; intro H'; insterU H'.
adamc@234 830
adamc@234 831 Section t6.
adamc@234 832 Variables A B : Type.
adamc@234 833 Variable P : A -> B -> Prop.
adamc@234 834 Variable f : A -> A -> A.
adamc@234 835 Variable g : B -> B -> B.
adamc@234 836
adamc@234 837 Hypothesis H1 : forall v, exists u, P v u.
adamc@234 838 Hypothesis H2 : forall v1 u1 v2 u2,
adamc@234 839 P v1 u1
adamc@234 840 -> P v2 u2
adamc@234 841 -> P (f v1 v2) (g u1 u2).
adamc@234 842
adamc@234 843 Theorem t6 : forall v1 v2, exists u1, exists u2, P (f v1 v2) (g u1 u2).
adamc@234 844 intros.
adamc@234 845
adam@328 846 (** 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@234 847
adamc@234 848 do 2 insterKeep H1.
adamc@234 849
adamc@234 850 (** Our proof state is extended with two generic instances of [H1].
adamc@234 851
adamc@234 852 [[
adamc@234 853 H' : exists u : B, P ?4289 u
adamc@234 854 H'0 : exists u : B, P ?4288 u
adamc@234 855 ============================
adamc@234 856 exists u1 : B, exists u2 : B, P (f v1 v2) (g u1 u2)
adamc@234 857
adamc@234 858 ]]
adamc@234 859
adam@328 860 [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@234 861
adamc@234 862 repeat match goal with
adamc@234 863 | [ H : ex _ |- _ ] => destruct H
adamc@234 864 end.
adamc@234 865
adamc@234 866 (** Now the goal is simple enough to solve by logic programming. *)
adamc@234 867
adamc@234 868 eauto.
adamc@234 869 Qed.
adamc@234 870 End t6.
adamc@234 871
adamc@234 872 (** 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@234 873
adamc@234 874 Section t7.
adamc@234 875 Variables A B : Type.
adamc@234 876 Variable Q : A -> Prop.
adamc@234 877 Variable P : A -> B -> Prop.
adamc@234 878 Variable f : A -> A -> A.
adamc@234 879 Variable g : B -> B -> B.
adamc@234 880
adamc@234 881 Hypothesis H1 : forall v, Q v -> exists u, P v u.
adamc@234 882 Hypothesis H2 : forall v1 u1 v2 u2,
adamc@234 883 P v1 u1
adamc@234 884 -> P v2 u2
adamc@234 885 -> P (f v1 v2) (g u1 u2).
adamc@234 886
adam@297 887 Theorem t7 : forall v1 v2, Q v1 -> Q v2 -> exists u1, exists u2, P (f v1 v2) (g u1 u2).
adamc@234 888 intros; do 2 insterKeep H1;
adamc@234 889 repeat match goal with
adamc@234 890 | [ H : ex _ |- _ ] => destruct H
adamc@234 891 end; eauto.
adamc@234 892
adamc@234 893 (** This proof script does not hit any errors until the very end, when an error message like this one is displayed.
adamc@234 894
adam@328 895 <<
adamc@234 896 No more subgoals but non-instantiated existential variables :
adamc@234 897 Existential 1 =
adam@328 898 >>
adam@328 899 [[
adamc@234 900 ?4384 : [A : Type
adamc@234 901 B : Type
adamc@234 902 Q : A -> Prop
adamc@234 903 P : A -> B -> Prop
adamc@234 904 f : A -> A -> A
adamc@234 905 g : B -> B -> B
adamc@234 906 H1 : forall v : A, Q v -> exists u : B, P v u
adamc@234 907 H2 : forall (v1 : A) (u1 : B) (v2 : A) (u2 : B),
adamc@234 908 P v1 u1 -> P v2 u2 -> P (f v1 v2) (g u1 u2)
adamc@234 909 v1 : A
adamc@234 910 v2 : A
adamc@234 911 H : Q v1
adamc@234 912 H0 : Q v2
adamc@234 913 H' : Q v2 -> exists u : B, P v2 u |- Q v2]
adamc@234 914
adamc@234 915 ]]
adamc@234 916
adam@288 917 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@234 918
adamc@234 919 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@234 920
adamc@234 921 Abort.
adamc@234 922 End t7.
adamc@234 923
adam@328 924 (* begin hide *)
adamc@234 925 Reset insterU.
adam@328 926 (* end hide *)
adam@328 927 (** %\noindent\coqdockw{%#<tt>#Reset#</tt>#%}% [insterU.] *)
adamc@234 928
adam@328 929 (** 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@234 930
adamc@234 931 Ltac insterU tac H :=
adamc@234 932 repeat match type of H with
adamc@234 933 | forall x : ?T, _ =>
adamc@234 934 match type of T with
adamc@234 935 | Prop =>
adamc@234 936 (let H' := fresh "H'" in
adam@328 937 assert (H' : T) by solve [ tac ];
adam@328 938 specialize (H H'); clear H')
adamc@234 939 || fail 1
adamc@234 940 | _ =>
adamc@234 941 let x := fresh "x" in
adamc@234 942 evar (x : T);
adam@328 943 let x' := eval unfold x in x in
adam@328 944 clear x; specialize (H x')
adamc@234 945 end
adamc@234 946 end.
adamc@234 947
adamc@234 948 Ltac insterKeep tac H :=
adamc@234 949 let H' := fresh "H'" in
adamc@234 950 generalize H; intro H'; insterU tac H'.
adamc@234 951
adamc@234 952 Section t7.
adamc@234 953 Variables A B : Type.
adamc@234 954 Variable Q : A -> Prop.
adamc@234 955 Variable P : A -> B -> Prop.
adamc@234 956 Variable f : A -> A -> A.
adamc@234 957 Variable g : B -> B -> B.
adamc@234 958
adamc@234 959 Hypothesis H1 : forall v, Q v -> exists u, P v u.
adamc@234 960 Hypothesis H2 : forall v1 u1 v2 u2,
adamc@234 961 P v1 u1
adamc@234 962 -> P v2 u2
adamc@234 963 -> P (f v1 v2) (g u1 u2).
adamc@234 964
adamc@234 965 Theorem t6 : forall v1 v2, Q v1 -> Q v2 -> exists u1, exists u2, P (f v1 v2) (g u1 u2).
adamc@234 966
adamc@234 967 (** 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@234 968
adamc@234 969 intros; do 2 insterKeep ltac:(idtac; match goal with
adamc@234 970 | [ H : Q ?v |- _ ] =>
adamc@234 971 match goal with
adamc@234 972 | [ _ : context[P v _] |- _ ] => fail 1
adamc@234 973 | _ => apply H
adamc@234 974 end
adamc@234 975 end) H1;
adamc@234 976 repeat match goal with
adamc@234 977 | [ H : ex _ |- _ ] => destruct H
adamc@234 978 end; eauto.
adamc@234 979 Qed.
adamc@234 980 End t7.
adamc@234 981
adamc@234 982 (** It is often useful to instantiate existential variables explicitly. A built-in tactic provides one way of doing so. *)
adamc@234 983
adamc@234 984 Theorem t8 : exists p : nat * nat, fst p = 3.
adamc@234 985 econstructor; instantiate (1 := (3, 2)); reflexivity.
adamc@234 986 Qed.
adamc@234 987
adamc@234 988 (** 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@234 989
adam@328 990 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@234 991
adamc@234 992 Ltac equate x y :=
adamc@234 993 let H := fresh "H" in
adam@328 994 assert (H : x = y) by reflexivity; clear H.
adamc@234 995
adam@328 996 (** 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@234 997
adamc@234 998 Theorem t9 : exists p : nat * nat, fst p = 3.
adamc@234 999 econstructor; match goal with
adamc@234 1000 | [ |- fst ?x = 3 ] => equate x (3, 2)
adamc@234 1001 end; reflexivity.
adamc@234 1002 Qed.