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Working with evars

author | Adam Chlipala <adamc@hcoop.net> |
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date | Mon, 30 Nov 2009 15:41:51 -0500 |

parents | 15501145d696 |

children | b653e6b19b6d |

rev | line source |
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adamc@220 | 1 (* Copyright (c) 2008-2009, 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 |

adamc@132 | 13 Require Import Tactics. |

adamc@132 | 14 |

adamc@132 | 15 Set Implicit Arguments. |

adamc@132 | 16 (* end hide *) |

adamc@132 | 17 |

adamc@132 | 18 |

adamc@132 | 19 (** %\part{Proof Engineering} |

adamc@132 | 20 |

adamc@132 | 21 \chapter{Proof Search in Ltac}% *) |

adamc@132 | 22 |

adamc@132 | 23 (** 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 | 24 |

adamc@132 | 25 (** * Some Built-In Automation Tactics *) |

adamc@132 | 26 |

adamc@132 | 27 (** 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 | 28 |

adamc@132 | 29 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 | 30 |

adamc@133 | 31 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 | 32 |

adamc@132 | 33 |

adamc@133 | 34 (** * Hint Databases *) |

adamc@132 | 35 |

adamc@133 | 36 (** Another class of built-in tactics includes [auto], [eauto], and [autorewrite]. These are based on %\textit{%#<i>#hint databases#</i>#%}%, which we have seen extended in many examples so far. These tactics are important, because, in Ltac programming, we cannot create "global variables" whose values can be extended seamlessly by different modules in different source files. We have seen the advantages of hints so far, where [crush] can be defined once and for all, while still automatically applying the hints we add throughout developments. |

adamc@133 | 37 |

adamc@220 | 38 The basic hints for [auto] and [eauto] are [Hint Immediate lemma], asking to try solving a goal immediately by applying a lemma and discharging any hypotheses with a single proof step each; [Resolve lemma], which does the same but may add new premises that are themselves to be subjects of nested proof search; [Constructor type], which acts like [Resolve] applied to every constructor of an inductive type; and [Unfold ident], which tries unfolding [ident] when it appears at the head of a proof goal. Each of these [Hint] commands may be used with a suffix, as in [Hint Resolve lemma : my_db]. This adds the hint only to the specified database, so that it would only be used by, for instance, [auto with my_db]. An additional argument to [auto] specifies the maximum depth of proof trees to search in depth-first order, as in [auto 8] or [auto 8 with my_db]. The default depth is 5. |

adamc@133 | 39 |

adamc@133 | 40 All of these [Hint] commands can be issued alternatively with a more primitive hint kind, [Extern]. A few examples should do best to explain how [Hint Extern] works. *) |

adamc@133 | 41 |

adamc@133 | 42 Theorem bool_neq : true <> false. |

adamc@141 | 43 (* begin thide *) |

adamc@133 | 44 auto. |

adamc@220 | 45 |

adamc@133 | 46 (** [crush] would have discharged this goal, but the default hint database for [auto] contains no hint that applies. *) |

adamc@220 | 47 |

adamc@133 | 48 Abort. |

adamc@133 | 49 |

adamc@133 | 50 (** It is hard to come up with a [bool]-specific hint that is not just a restatement of the theorem we mean to prove. Luckily, a simpler form suffices. *) |

adamc@133 | 51 |

adamc@133 | 52 Hint Extern 1 (_ <> _) => congruence. |

adamc@133 | 53 |

adamc@133 | 54 Theorem bool_neq : true <> false. |

adamc@133 | 55 auto. |

adamc@133 | 56 Qed. |

adamc@141 | 57 (* end thide *) |

adamc@133 | 58 |

adamc@133 | 59 (** Our hint says: "whenever the conclusion matches the pattern [_ <> _], try applying [congruence]." The [1] is a cost for this rule. During proof search, whenever multiple rules apply, rules are tried in increasing cost order, so it pays to assign high costs to relatively expensive [Extern] hints. |

adamc@133 | 60 |

adamc@133 | 61 [Extern] hints may be implemented with the full Ltac language. This example shows a case where a hint uses a [match]. *) |

adamc@133 | 62 |

adamc@133 | 63 Section forall_and. |

adamc@133 | 64 Variable A : Set. |

adamc@133 | 65 Variables P Q : A -> Prop. |

adamc@133 | 66 |

adamc@133 | 67 Hypothesis both : forall x, P x /\ Q x. |

adamc@133 | 68 |

adamc@133 | 69 Theorem forall_and : forall z, P z. |

adamc@141 | 70 (* begin thide *) |

adamc@133 | 71 crush. |

adamc@220 | 72 |

adamc@133 | 73 (** [crush] makes no progress beyond what [intros] would have accomplished. [auto] will not apply the hypothesis [both] to prove the goal, because the conclusion of [both] does not unify with the conclusion of the goal. However, we can teach [auto] to handle this kind of goal. *) |

adamc@133 | 74 |

adamc@133 | 75 Hint Extern 1 (P ?X) => |

adamc@133 | 76 match goal with |

adamc@133 | 77 | [ H : forall x, P x /\ _ |- _ ] => apply (proj1 (H X)) |

adamc@133 | 78 end. |

adamc@133 | 79 |

adamc@133 | 80 auto. |

adamc@133 | 81 Qed. |

adamc@141 | 82 (* end thide *) |

adamc@133 | 83 |

adamc@133 | 84 (** We see that an [Extern] pattern may bind unification variables that we use in the associated tactic. [proj1] is a function from the standard library for extracting a proof of [R] from a proof of [R /\ S]. *) |

adamc@220 | 85 |

adamc@133 | 86 End forall_and. |

adamc@133 | 87 |

adamc@133 | 88 (** After our success on this example, we might get more ambitious and seek to generalize the hint to all possible predicates [P]. |

adamc@133 | 89 |

adamc@133 | 90 [[ |

adamc@133 | 91 Hint Extern 1 (?P ?X) => |

adamc@133 | 92 match goal with |

adamc@220 | 93 | [ H : forall x, P x /\ _ |- _ ] => apply (proj1 (H X)) |

adamc@133 | 94 end. |

adamc@133 | 95 |

adamc@133 | 96 User error: Bound head variable |

adamc@220 | 97 |

adamc@134 | 98 ]] |

adamc@133 | 99 |

adamc@134 | 100 Coq's [auto] hint databases work as tables mapping %\textit{%#<i>#head symbols#</i>#%}% to lists of tactics to try. Because of this, the constant head of an [Extern] pattern must be determinable statically. In our first [Extern] hint, the head symbol was [not], since [x <> y] desugars to [not (eq x y)]; and, in the second example, the head symbol was [P]. |

adamc@133 | 101 |

adamc@134 | 102 This restriction on [Extern] hints is the main limitation of the [auto] mechanism, preventing us from using it for general context simplifications that are not keyed off of the form of the conclusion. This is perhaps just as well, since we can often code more efficient tactics with specialized Ltac programs, and we will see how in later sections of the chapter. |

adamc@134 | 103 |

adamc@134 | 104 We have used [Hint Rewrite] in many examples so far. [crush] uses these hints by calling [autorewrite]. Our rewrite hints have taken the form [Hint Rewrite lemma : cpdt], adding them to the [cpdt] rewrite database. This is because, in contrast to [auto], [autorewrite] has no default database. Thus, we set the convention that [crush] uses the [cpdt] database. |

adamc@134 | 105 |

adamc@134 | 106 This example shows a direct use of [autorewrite]. *) |

adamc@134 | 107 |

adamc@134 | 108 Section autorewrite. |

adamc@134 | 109 Variable A : Set. |

adamc@134 | 110 Variable f : A -> A. |

adamc@134 | 111 |

adamc@134 | 112 Hypothesis f_f : forall x, f (f x) = f x. |

adamc@134 | 113 |

adamc@134 | 114 Hint Rewrite f_f : my_db. |

adamc@134 | 115 |

adamc@134 | 116 Lemma f_f_f : forall x, f (f (f x)) = f x. |

adamc@134 | 117 intros; autorewrite with my_db; reflexivity. |

adamc@134 | 118 Qed. |

adamc@134 | 119 |

adamc@134 | 120 (** There are a few ways in which [autorewrite] can lead to trouble when insufficient care is taken in choosing hints. First, the set of hints may define a nonterminating rewrite system, in which case invocations to [autorewrite] may not terminate. Second, we may add hints that "lead [autorewrite] down the wrong path." For instance: *) |

adamc@134 | 121 |

adamc@134 | 122 Section garden_path. |

adamc@134 | 123 Variable g : A -> A. |

adamc@134 | 124 Hypothesis f_g : forall x, f x = g x. |

adamc@134 | 125 Hint Rewrite f_g : my_db. |

adamc@134 | 126 |

adamc@134 | 127 Lemma f_f_f' : forall x, f (f (f x)) = f x. |

adamc@134 | 128 intros; autorewrite with my_db. |

adamc@134 | 129 (** [[ |

adamc@134 | 130 ============================ |

adamc@134 | 131 g (g (g x)) = g x |

adamc@134 | 132 ]] *) |

adamc@220 | 133 |

adamc@134 | 134 Abort. |

adamc@134 | 135 |

adamc@134 | 136 (** Our new hint was used to rewrite the goal into a form where the old hint could no longer be applied. This "non-monotonicity" of rewrite hints contrasts with the situation for [auto], where new hints may slow down proof search but can never "break" old proofs. *) |

adamc@134 | 137 |

adamc@134 | 138 Reset garden_path. |

adamc@134 | 139 |

adamc@134 | 140 (** [autorewrite] works with quantified equalities that include additional premises, but we must be careful to avoid similar incorrect rewritings. *) |

adamc@134 | 141 |

adamc@134 | 142 Section garden_path. |

adamc@134 | 143 Variable P : A -> Prop. |

adamc@134 | 144 Variable g : A -> A. |

adamc@134 | 145 Hypothesis f_g : forall x, P x -> f x = g x. |

adamc@134 | 146 Hint Rewrite f_g : my_db. |

adamc@134 | 147 |

adamc@134 | 148 Lemma f_f_f' : forall x, f (f (f x)) = f x. |

adamc@134 | 149 intros; autorewrite with my_db. |

adamc@134 | 150 (** [[ |

adamc@134 | 151 |

adamc@134 | 152 ============================ |

adamc@134 | 153 g (g (g x)) = g x |

adamc@134 | 154 |

adamc@134 | 155 subgoal 2 is: |

adamc@134 | 156 P x |

adamc@134 | 157 subgoal 3 is: |

adamc@134 | 158 P (f x) |

adamc@134 | 159 subgoal 4 is: |

adamc@134 | 160 P (f x) |

adamc@134 | 161 ]] *) |

adamc@220 | 162 |

adamc@134 | 163 Abort. |

adamc@134 | 164 |

adamc@134 | 165 (** The inappropriate rule fired the same three times as before, even though we know we will not be able to prove the premises. *) |

adamc@134 | 166 |

adamc@134 | 167 Reset garden_path. |

adamc@134 | 168 |

adamc@134 | 169 (** Our final, successful, attempt uses an extra argument to [Hint Rewrite] that specifies a tactic to apply to generated premises. *) |

adamc@134 | 170 |

adamc@134 | 171 Section garden_path. |

adamc@134 | 172 Variable P : A -> Prop. |

adamc@134 | 173 Variable g : A -> A. |

adamc@134 | 174 Hypothesis f_g : forall x, P x -> f x = g x. |

adamc@141 | 175 (* begin thide *) |

adamc@134 | 176 Hint Rewrite f_g using assumption : my_db. |

adamc@141 | 177 (* end thide *) |

adamc@134 | 178 |

adamc@134 | 179 Lemma f_f_f' : forall x, f (f (f x)) = f x. |

adamc@141 | 180 (* begin thide *) |

adamc@134 | 181 intros; autorewrite with my_db; reflexivity. |

adamc@134 | 182 Qed. |

adamc@141 | 183 (* end thide *) |

adamc@134 | 184 |

adamc@134 | 185 (** [autorewrite] will still use [f_g] when the generated premise is among our assumptions. *) |

adamc@134 | 186 |

adamc@134 | 187 Lemma f_f_f_g : forall x, P x -> f (f x) = g x. |

adamc@141 | 188 (* begin thide *) |

adamc@134 | 189 intros; autorewrite with my_db; reflexivity. |

adamc@141 | 190 (* end thide *) |

adamc@134 | 191 Qed. |

adamc@134 | 192 End garden_path. |

adamc@134 | 193 |

adamc@220 | 194 (** remove printing * *) |

adamc@220 | 195 |

adamc@134 | 196 (** It can also be useful to use the [autorewrite with db in *] form, which does rewriting in hypotheses, as well as in the conclusion. *) |

adamc@134 | 197 |

adamc@220 | 198 (** printing * $*$ *) |

adamc@220 | 199 |

adamc@134 | 200 Lemma in_star : forall x y, f (f (f (f x))) = f (f y) |

adamc@134 | 201 -> f x = f (f (f y)). |

adamc@141 | 202 (* begin thide *) |

adamc@134 | 203 intros; autorewrite with my_db in *; assumption. |

adamc@141 | 204 (* end thide *) |

adamc@134 | 205 Qed. |

adamc@134 | 206 |

adamc@134 | 207 End autorewrite. |

adamc@135 | 208 |

adamc@135 | 209 |

adamc@135 | 210 (** * Ltac Programming Basics *) |

adamc@135 | 211 |

adamc@135 | 212 (** 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 | 213 |

adamc@135 | 214 One common use for [match] tactics is identification of subjects for case analysis, as we see in this tactic definition. *) |

adamc@135 | 215 |

adamc@141 | 216 (* begin thide *) |

adamc@135 | 217 Ltac find_if := |

adamc@135 | 218 match goal with |

adamc@135 | 219 | [ |- if ?X then _ else _ ] => destruct X |

adamc@135 | 220 end. |

adamc@141 | 221 (* end thide *) |

adamc@135 | 222 |

adamc@135 | 223 (** 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 | 224 |

adamc@135 | 225 Theorem hmm : forall (a b c : bool), |

adamc@135 | 226 if a |

adamc@135 | 227 then if b |

adamc@135 | 228 then True |

adamc@135 | 229 else True |

adamc@135 | 230 else if c |

adamc@135 | 231 then True |

adamc@135 | 232 else True. |

adamc@141 | 233 (* begin thide *) |

adamc@135 | 234 intros; repeat find_if; constructor. |

adamc@135 | 235 Qed. |

adamc@141 | 236 (* end thide *) |

adamc@135 | 237 |

adamc@135 | 238 (** 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 | 239 |

adamc@135 | 240 Another very useful Ltac building block is %\textit{%#<i>#context patterns#</i>#%}%. *) |

adamc@135 | 241 |

adamc@141 | 242 (* begin thide *) |

adamc@135 | 243 Ltac find_if_inside := |

adamc@135 | 244 match goal with |

adamc@135 | 245 | [ |- context[if ?X then _ else _] ] => destruct X |

adamc@135 | 246 end. |

adamc@141 | 247 (* end thide *) |

adamc@135 | 248 |

adamc@135 | 249 (** 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 | 250 |

adamc@135 | 251 Theorem hmm' : forall (a b c : bool), |

adamc@135 | 252 if a |

adamc@135 | 253 then if b |

adamc@135 | 254 then True |

adamc@135 | 255 else True |

adamc@135 | 256 else if c |

adamc@135 | 257 then True |

adamc@135 | 258 else True. |

adamc@141 | 259 (* begin thide *) |

adamc@135 | 260 intros; repeat find_if_inside; constructor. |

adamc@135 | 261 Qed. |

adamc@141 | 262 (* end thide *) |

adamc@135 | 263 |

adamc@135 | 264 (** We can also use [find_if_inside] to prove goals that [find_if] does not simplify sufficiently. *) |

adamc@135 | 265 |

adamc@141 | 266 Theorem hmm2 : forall (a b : bool), |

adamc@135 | 267 (if a then 42 else 42) = (if b then 42 else 42). |

adamc@141 | 268 (* begin thide *) |

adamc@135 | 269 intros; repeat find_if_inside; reflexivity. |

adamc@135 | 270 Qed. |

adamc@141 | 271 (* end thide *) |

adamc@135 | 272 |

adamc@135 | 273 (** 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 | 274 |

adamc@141 | 275 (* begin thide *) |

adamc@135 | 276 Ltac my_tauto := |

adamc@135 | 277 repeat match goal with |

adamc@135 | 278 | [ H : ?P |- ?P ] => exact H |

adamc@135 | 279 |

adamc@135 | 280 | [ |- True ] => constructor |

adamc@135 | 281 | [ |- _ /\ _ ] => constructor |

adamc@135 | 282 | [ |- _ -> _ ] => intro |

adamc@135 | 283 |

adamc@135 | 284 | [ H : False |- _ ] => destruct H |

adamc@135 | 285 | [ H : _ /\ _ |- _ ] => destruct H |

adamc@135 | 286 | [ H : _ \/ _ |- _ ] => destruct H |

adamc@135 | 287 |

adamc@135 | 288 | [ H1 : ?P -> ?Q, H2 : ?P |- _ ] => |

adamc@135 | 289 let H := fresh "H" in |

adamc@135 | 290 generalize (H1 H2); clear H1; intro H |

adamc@135 | 291 end. |

adamc@141 | 292 (* end thide *) |

adamc@135 | 293 |

adamc@135 | 294 (** 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 | 295 |

adamc@220 | 296 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 | 297 |

adamc@135 | 298 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 | 299 |

adamc@135 | 300 Section propositional. |

adamc@135 | 301 Variables P Q R : Prop. |

adamc@135 | 302 |

adamc@138 | 303 Theorem propositional : (P \/ Q \/ False) /\ (P -> Q) -> True /\ Q. |

adamc@141 | 304 (* begin thide *) |

adamc@135 | 305 my_tauto. |

adamc@135 | 306 Qed. |

adamc@141 | 307 (* end thide *) |

adamc@135 | 308 End propositional. |

adamc@135 | 309 |

adamc@135 | 310 (** 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 | 311 |

adamc@135 | 312 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 | 313 |

adamc@135 | 314 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 | 315 |

adamc@135 | 316 For instance, this (unnecessarily verbose) proof script works: *) |

adamc@135 | 317 |

adamc@135 | 318 Theorem m1 : True. |

adamc@135 | 319 match goal with |

adamc@135 | 320 | [ |- _ ] => intro |

adamc@135 | 321 | [ |- True ] => constructor |

adamc@135 | 322 end. |

adamc@141 | 323 (* begin thide *) |

adamc@135 | 324 Qed. |

adamc@141 | 325 (* end thide *) |

adamc@135 | 326 |

adamc@135 | 327 (** 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 | 328 |

adamc@135 | 329 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 | 330 |

adamc@135 | 331 Theorem m2 : forall P Q R : Prop, P -> Q -> R -> Q. |

adamc@135 | 332 intros; match goal with |

adamc@220 | 333 | [ H : _ |- _ ] => idtac H |

adamc@135 | 334 end. |

adamc@135 | 335 |

adamc@220 | 336 (** 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 | 337 |

adamc@141 | 338 (* begin thide *) |

adamc@135 | 339 match goal with |

adamc@135 | 340 | [ H : _ |- _ ] => exact H |

adamc@135 | 341 end. |

adamc@135 | 342 Qed. |

adamc@141 | 343 (* end thide *) |

adamc@135 | 344 |

adamc@135 | 345 (** 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 | 346 |

adamc@135 | 347 (** Now we are equipped to implement a tactic for checking that a proposition is not among our hypotheses: *) |

adamc@135 | 348 |

adamc@141 | 349 (* begin thide *) |

adamc@135 | 350 Ltac notHyp P := |

adamc@135 | 351 match goal with |

adamc@135 | 352 | [ _ : P |- _ ] => fail 1 |

adamc@135 | 353 | _ => |

adamc@135 | 354 match P with |

adamc@135 | 355 | ?P1 /\ ?P2 => first [ notHyp P1 | notHyp P2 | fail 2 ] |

adamc@135 | 356 | _ => idtac |

adamc@135 | 357 end |

adamc@135 | 358 end. |

adamc@141 | 359 (* end thide *) |

adamc@135 | 360 |

adamc@135 | 361 (** 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 | 362 |

adamc@135 | 363 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 | 364 |

adamc@135 | 365 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 | 366 |

adamc@135 | 367 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 | 368 |

adamc@141 | 369 (* begin thide *) |

adamc@135 | 370 Ltac extend pf := |

adamc@135 | 371 let t := type of pf in |

adamc@135 | 372 notHyp t; generalize pf; intro. |

adamc@141 | 373 (* end thide *) |

adamc@135 | 374 |

adamc@135 | 375 (** 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 | 376 |

adamc@135 | 377 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 | 378 |

adamc@141 | 379 (* begin thide *) |

adamc@135 | 380 Ltac completer := |

adamc@135 | 381 repeat match goal with |

adamc@135 | 382 | [ |- _ /\ _ ] => constructor |

adamc@135 | 383 | [ H : _ /\ _ |- _ ] => destruct H |

adamc@135 | 384 | [ H : ?P -> ?Q, H' : ?P |- _ ] => |

adamc@135 | 385 generalize (H H'); clear H; intro H |

adamc@135 | 386 | [ |- forall x, _ ] => intro |

adamc@135 | 387 |

adamc@135 | 388 | [ H : forall x, ?P x -> _, H' : ?P ?X |- _ ] => |

adamc@135 | 389 extend (H X H') |

adamc@135 | 390 end. |

adamc@141 | 391 (* end thide *) |

adamc@135 | 392 |

adamc@135 | 393 (** 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 | 394 |

adamc@135 | 395 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 | 396 |

adamc@135 | 397 We can check that [completer] is working properly: *) |

adamc@135 | 398 |

adamc@135 | 399 Section firstorder. |

adamc@135 | 400 Variable A : Set. |

adamc@135 | 401 Variables P Q R S : A -> Prop. |

adamc@135 | 402 |

adamc@135 | 403 Hypothesis H1 : forall x, P x -> Q x /\ R x. |

adamc@135 | 404 Hypothesis H2 : forall x, R x -> S x. |

adamc@135 | 405 |

adamc@135 | 406 Theorem fo : forall x, P x -> S x. |

adamc@141 | 407 (* begin thide *) |

adamc@135 | 408 completer. |

adamc@135 | 409 (** [[ |

adamc@135 | 410 x : A |

adamc@135 | 411 H : P x |

adamc@135 | 412 H0 : Q x |

adamc@135 | 413 H3 : R x |

adamc@135 | 414 H4 : S x |

adamc@135 | 415 ============================ |

adamc@135 | 416 S x |

adamc@135 | 417 ]] *) |

adamc@135 | 418 |

adamc@135 | 419 assumption. |

adamc@135 | 420 Qed. |

adamc@141 | 421 (* end thide *) |

adamc@135 | 422 End firstorder. |

adamc@135 | 423 |

adamc@135 | 424 (** 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 | 425 |

adamc@141 | 426 (* begin thide *) |

adamc@135 | 427 Ltac completer' := |

adamc@135 | 428 repeat match goal with |

adamc@135 | 429 | [ |- _ /\ _ ] => constructor |

adamc@135 | 430 | [ H : _ /\ _ |- _ ] => destruct H |

adamc@135 | 431 | [ H : ?P -> _, H' : ?P |- _ ] => |

adamc@135 | 432 generalize (H H'); clear H; intro H |

adamc@135 | 433 | [ |- forall x, _ ] => intro |

adamc@135 | 434 |

adamc@135 | 435 | [ H : forall x, ?P x -> _, H' : ?P ?X |- _ ] => |

adamc@135 | 436 extend (H X H') |

adamc@135 | 437 end. |

adamc@141 | 438 (* end thide *) |

adamc@135 | 439 |

adamc@135 | 440 (** 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 | 441 |

adamc@135 | 442 Section firstorder'. |

adamc@135 | 443 Variable A : Set. |

adamc@135 | 444 Variables P Q R S : A -> Prop. |

adamc@135 | 445 |

adamc@135 | 446 Hypothesis H1 : forall x, P x -> Q x /\ R x. |

adamc@135 | 447 Hypothesis H2 : forall x, R x -> S x. |

adamc@135 | 448 |

adamc@135 | 449 Theorem fo' : forall x, P x -> S x. |

adamc@141 | 450 (* begin thide *) |

adamc@135 | 451 (** [[ |

adamc@135 | 452 completer'. |

adamc@220 | 453 |

adamc@205 | 454 ]] |

adamc@205 | 455 |

adamc@135 | 456 Coq loops forever at this point. What went wrong? *) |

adamc@220 | 457 |

adamc@135 | 458 Abort. |

adamc@141 | 459 (* end thide *) |

adamc@135 | 460 End firstorder'. |

adamc@136 | 461 |

adamc@136 | 462 (** A few examples should illustrate the issue. Here we see a [match]-based proof that works fine: *) |

adamc@136 | 463 |

adamc@136 | 464 Theorem t1 : forall x : nat, x = x. |

adamc@136 | 465 match goal with |

adamc@136 | 466 | [ |- forall x, _ ] => trivial |

adamc@136 | 467 end. |

adamc@141 | 468 (* begin thide *) |

adamc@136 | 469 Qed. |

adamc@141 | 470 (* end thide *) |

adamc@136 | 471 |

adamc@136 | 472 (** This one fails. *) |

adamc@136 | 473 |

adamc@141 | 474 (* begin thide *) |

adamc@136 | 475 Theorem t1' : forall x : nat, x = x. |

adamc@136 | 476 (** [[ |

adamc@136 | 477 match goal with |

adamc@136 | 478 | [ |- forall x, ?P ] => trivial |

adamc@136 | 479 end. |

adamc@136 | 480 |

adamc@136 | 481 User error: No matching clauses for match goal |

adamc@136 | 482 ]] *) |

adamc@220 | 483 |

adamc@136 | 484 Abort. |

adamc@141 | 485 (* end thide *) |

adamc@136 | 486 |

adamc@136 | 487 (** 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. |

adamc@136 | 488 |

adamc@136 | 489 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 | 490 |

adamc@136 | 491 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. *) |

adamc@137 | 492 |

adamc@137 | 493 |

adamc@137 | 494 (** * Functional Programming in Ltac *) |

adamc@137 | 495 |

adamc@141 | 496 (* EX: Write a list length function in Ltac. *) |

adamc@141 | 497 |

adamc@137 | 498 (** 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 | 499 |

adamc@137 | 500 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 | 501 |

adamc@137 | 502 [[ |

adamc@137 | 503 Ltac length ls := |

adamc@137 | 504 match ls with |

adamc@137 | 505 | nil => O |

adamc@137 | 506 | _ :: ls' => S (length ls') |

adamc@137 | 507 end. |

adamc@137 | 508 |

adamc@137 | 509 Error: The reference ls' was not found in the current environment |

adamc@220 | 510 |

adamc@137 | 511 ]] |

adamc@137 | 512 |

adamc@137 | 513 At this point, we hopefully remember that pattern variable names must be prefixed by question marks in Ltac. |

adamc@137 | 514 |

adamc@137 | 515 [[ |

adamc@137 | 516 Ltac length ls := |

adamc@137 | 517 match ls with |

adamc@137 | 518 | nil => O |

adamc@137 | 519 | _ :: ?ls' => S (length ls') |

adamc@137 | 520 end. |

adamc@137 | 521 |

adamc@137 | 522 Error: The reference S was not found in the current environment |

adamc@220 | 523 |

adamc@137 | 524 ]] |

adamc@137 | 525 |

adamc@137 | 526 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 | 527 |

adamc@141 | 528 (* begin thide *) |

adamc@137 | 529 Ltac length ls := |

adamc@137 | 530 match ls with |

adamc@137 | 531 | nil => O |

adamc@137 | 532 | _ :: ?ls' => constr:(S (length ls')) |

adamc@137 | 533 end. |

adamc@137 | 534 |

adamc@137 | 535 (** This definition is accepted. It can be a little awkward to test Ltac definitions like this. Here is one method. *) |

adamc@137 | 536 |

adamc@137 | 537 Goal False. |

adamc@137 | 538 let n := length (1 :: 2 :: 3 :: nil) in |

adamc@137 | 539 pose n. |

adamc@137 | 540 (** [[ |

adamc@137 | 541 n := S (length (2 :: 3 :: nil)) : nat |

adamc@137 | 542 ============================ |

adamc@137 | 543 False |

adamc@220 | 544 |

adamc@137 | 545 ]] |

adamc@137 | 546 |

adamc@220 | 547 We use the [pose] tactic, which extends the proof context with a new variable that is set equal to particular a 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 | 548 |

adamc@220 | 549 [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 | 550 |

adamc@220 | 551 Abort. |

adamc@137 | 552 |

adamc@137 | 553 Reset length. |

adamc@137 | 554 |

adamc@137 | 555 (** 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 | 556 |

adamc@137 | 557 Ltac length ls := |

adamc@137 | 558 match ls with |

adamc@137 | 559 | nil => O |

adamc@137 | 560 | _ :: ?ls' => |

adamc@137 | 561 let ls'' := length ls' in |

adamc@137 | 562 constr:(S ls'') |

adamc@137 | 563 end. |

adamc@137 | 564 |

adamc@137 | 565 Goal False. |

adamc@137 | 566 let n := length (1 :: 2 :: 3 :: nil) in |

adamc@137 | 567 pose n. |

adamc@137 | 568 (** [[ |

adamc@137 | 569 n := 3 : nat |

adamc@137 | 570 ============================ |

adamc@137 | 571 False |

adamc@137 | 572 ]] *) |

adamc@220 | 573 |

adamc@137 | 574 Abort. |

adamc@141 | 575 (* end thide *) |

adamc@141 | 576 |

adamc@141 | 577 (* EX: Write a list map function in Ltac. *) |

adamc@137 | 578 |

adamc@137 | 579 (** 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 | 580 |

adamc@141 | 581 (* begin thide *) |

adamc@137 | 582 Ltac map T f := |

adamc@137 | 583 let rec map' ls := |

adamc@137 | 584 match ls with |

adamc@137 | 585 | nil => constr:(@nil T) |

adamc@137 | 586 | ?x :: ?ls' => |

adamc@137 | 587 let x' := f x in |

adamc@137 | 588 let ls'' := map' ls' in |

adamc@137 | 589 constr:(x' :: ls'') |

adamc@137 | 590 end in |

adamc@137 | 591 map'. |

adamc@137 | 592 |

adamc@137 | 593 (** 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 | 594 |

adamc@137 | 595 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 | 596 |

adamc@137 | 597 Goal False. |

adamc@137 | 598 let ls := map (nat * nat)%type ltac:(fun x => constr:(x, x)) (1 :: 2 :: 3 :: nil) in |

adamc@137 | 599 pose ls. |

adamc@137 | 600 (** [[ |

adamc@137 | 601 l := (1, 1) :: (2, 2) :: (3, 3) :: nil : list (nat * nat) |

adamc@137 | 602 ============================ |

adamc@137 | 603 False |

adamc@137 | 604 ]] *) |

adamc@220 | 605 |

adamc@137 | 606 Abort. |

adamc@141 | 607 (* end thide *) |

adamc@137 | 608 |

adamc@138 | 609 |

adamc@139 | 610 (** * Recursive Proof Search *) |

adamc@139 | 611 |

adamc@139 | 612 (** 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 | 613 |

adamc@139 | 614 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 | 615 |

adamc@141 | 616 (* begin thide *) |

adamc@139 | 617 Ltac inster n := |

adamc@139 | 618 intuition; |

adamc@139 | 619 match n with |

adamc@139 | 620 | S ?n' => |

adamc@139 | 621 match goal with |

adamc@139 | 622 | [ H : forall x : ?T, _, x : ?T |- _ ] => generalize (H x); inster n' |

adamc@139 | 623 end |

adamc@139 | 624 end. |

adamc@141 | 625 (* end thide *) |

adamc@139 | 626 |

adamc@139 | 627 (** [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 | 628 |

adamc@139 | 629 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 | 630 |

adamc@139 | 631 Section test_inster. |

adamc@139 | 632 Variable A : Set. |

adamc@139 | 633 Variables P Q : A -> Prop. |

adamc@139 | 634 Variable f : A -> A. |

adamc@139 | 635 Variable g : A -> A -> A. |

adamc@139 | 636 |

adamc@139 | 637 Hypothesis H1 : forall x y, P (g x y) -> Q (f x). |

adamc@139 | 638 |

adamc@139 | 639 Theorem test_inster : forall x y, P (g x y) -> Q (f x). |

adamc@220 | 640 inster 2. |

adamc@139 | 641 Qed. |

adamc@139 | 642 |

adamc@139 | 643 Hypothesis H3 : forall u v, P u /\ P v /\ u <> v -> P (g u v). |

adamc@139 | 644 Hypothesis H4 : forall u, Q (f u) -> P u /\ P (f u). |

adamc@139 | 645 |

adamc@139 | 646 Theorem test_inster2 : forall x y, x <> y -> P x -> Q (f y) -> Q (f x). |

adamc@220 | 647 inster 3. |

adamc@139 | 648 Qed. |

adamc@139 | 649 End test_inster. |

adamc@139 | 650 |

adamc@140 | 651 (** 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 | 652 |

adamc@140 | 653 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 | 654 |

adamc@140 | 655 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 | 656 |

adamc@140 | 657 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 | 658 |

adamc@138 | 659 Definition imp (P1 P2 : Prop) := P1 -> P2. |

adamc@140 | 660 Infix "-->" := imp (no associativity, at level 95). |

adamc@140 | 661 Ltac imp := unfold imp; firstorder. |

adamc@138 | 662 |

adamc@140 | 663 (** These lemmas about [imp] will be useful in the tactic that we will write. *) |

adamc@138 | 664 |

adamc@138 | 665 Theorem and_True_prem : forall P Q, |

adamc@138 | 666 (P /\ True --> Q) |

adamc@138 | 667 -> (P --> Q). |

adamc@138 | 668 imp. |

adamc@138 | 669 Qed. |

adamc@138 | 670 |

adamc@138 | 671 Theorem and_True_conc : forall P Q, |

adamc@138 | 672 (P --> Q /\ True) |

adamc@138 | 673 -> (P --> Q). |

adamc@138 | 674 imp. |

adamc@138 | 675 Qed. |

adamc@138 | 676 |

adamc@138 | 677 Theorem assoc_prem1 : forall P Q R S, |

adamc@138 | 678 (P /\ (Q /\ R) --> S) |

adamc@138 | 679 -> ((P /\ Q) /\ R --> S). |

adamc@138 | 680 imp. |

adamc@138 | 681 Qed. |

adamc@138 | 682 |

adamc@138 | 683 Theorem assoc_prem2 : forall P Q R S, |

adamc@138 | 684 (Q /\ (P /\ R) --> S) |

adamc@138 | 685 -> ((P /\ Q) /\ R --> S). |

adamc@138 | 686 imp. |

adamc@138 | 687 Qed. |

adamc@138 | 688 |

adamc@138 | 689 Theorem comm_prem : forall P Q R, |

adamc@138 | 690 (P /\ Q --> R) |

adamc@138 | 691 -> (Q /\ P --> R). |

adamc@138 | 692 imp. |

adamc@138 | 693 Qed. |

adamc@138 | 694 |

adamc@138 | 695 Theorem assoc_conc1 : forall P Q R S, |

adamc@138 | 696 (S --> P /\ (Q /\ R)) |

adamc@138 | 697 -> (S --> (P /\ Q) /\ R). |

adamc@138 | 698 imp. |

adamc@138 | 699 Qed. |

adamc@138 | 700 |

adamc@138 | 701 Theorem assoc_conc2 : forall P Q R S, |

adamc@138 | 702 (S --> Q /\ (P /\ R)) |

adamc@138 | 703 -> (S --> (P /\ Q) /\ R). |

adamc@138 | 704 imp. |

adamc@138 | 705 Qed. |

adamc@138 | 706 |

adamc@138 | 707 Theorem comm_conc : forall P Q R, |

adamc@138 | 708 (R --> P /\ Q) |

adamc@138 | 709 -> (R --> Q /\ P). |

adamc@138 | 710 imp. |

adamc@138 | 711 Qed. |

adamc@138 | 712 |

adamc@140 | 713 (** 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 | 714 |

adamc@141 | 715 (* begin thide *) |

adamc@138 | 716 Ltac search_prem tac := |

adamc@138 | 717 let rec search P := |

adamc@138 | 718 tac |

adamc@138 | 719 || (apply and_True_prem; tac) |

adamc@138 | 720 || match P with |

adamc@138 | 721 | ?P1 /\ ?P2 => |

adamc@138 | 722 (apply assoc_prem1; search P1) |

adamc@138 | 723 || (apply assoc_prem2; search P2) |

adamc@138 | 724 end |

adamc@138 | 725 in match goal with |

adamc@138 | 726 | [ |- ?P /\ _ --> _ ] => search P |

adamc@138 | 727 | [ |- _ /\ ?P --> _ ] => apply comm_prem; search P |

adamc@138 | 728 | [ |- _ --> _ ] => progress (tac || (apply and_True_prem; tac)) |

adamc@138 | 729 end. |

adamc@138 | 730 |

adamc@140 | 731 (** 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 | 732 |

adamc@140 | 733 [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 | 734 |

adamc@140 | 735 We will also want a dual function [search_conc], which does tree search through an [imp] conclusion. *) |

adamc@140 | 736 |

adamc@138 | 737 Ltac search_conc tac := |

adamc@138 | 738 let rec search P := |

adamc@138 | 739 tac |

adamc@138 | 740 || (apply and_True_conc; tac) |

adamc@138 | 741 || match P with |

adamc@138 | 742 | ?P1 /\ ?P2 => |

adamc@138 | 743 (apply assoc_conc1; search P1) |

adamc@138 | 744 || (apply assoc_conc2; search P2) |

adamc@138 | 745 end |

adamc@138 | 746 in match goal with |

adamc@138 | 747 | [ |- _ --> ?P /\ _ ] => search P |

adamc@138 | 748 | [ |- _ --> _ /\ ?P ] => apply comm_conc; search P |

adamc@138 | 749 | [ |- _ --> _ ] => progress (tac || (apply and_True_conc; tac)) |

adamc@138 | 750 end. |

adamc@138 | 751 |

adamc@140 | 752 (** 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 | 753 |

adamc@138 | 754 Theorem False_prem : forall P Q, |

adamc@138 | 755 False /\ P --> Q. |

adamc@138 | 756 imp. |

adamc@138 | 757 Qed. |

adamc@138 | 758 |

adamc@138 | 759 Theorem True_conc : forall P Q : Prop, |

adamc@138 | 760 (P --> Q) |

adamc@138 | 761 -> (P --> True /\ Q). |

adamc@138 | 762 imp. |

adamc@138 | 763 Qed. |

adamc@138 | 764 |

adamc@138 | 765 Theorem Match : forall P Q R : Prop, |

adamc@138 | 766 (Q --> R) |

adamc@138 | 767 -> (P /\ Q --> P /\ R). |

adamc@138 | 768 imp. |

adamc@138 | 769 Qed. |

adamc@138 | 770 |

adamc@138 | 771 Theorem ex_prem : forall (T : Type) (P : T -> Prop) (Q R : Prop), |

adamc@138 | 772 (forall x, P x /\ Q --> R) |

adamc@138 | 773 -> (ex P /\ Q --> R). |

adamc@138 | 774 imp. |

adamc@138 | 775 Qed. |

adamc@138 | 776 |

adamc@138 | 777 Theorem ex_conc : forall (T : Type) (P : T -> Prop) (Q R : Prop) x, |

adamc@138 | 778 (Q --> P x /\ R) |

adamc@138 | 779 -> (Q --> ex P /\ R). |

adamc@138 | 780 imp. |

adamc@138 | 781 Qed. |

adamc@138 | 782 |

adamc@140 | 783 (** We will also want a "base case" lemma for finishing proofs where cancelation has removed every constituent of the conclusion. *) |

adamc@140 | 784 |

adamc@138 | 785 Theorem imp_True : forall P, |

adamc@138 | 786 P --> True. |

adamc@138 | 787 imp. |

adamc@138 | 788 Qed. |

adamc@138 | 789 |

adamc@220 | 790 (** 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 | 791 |

adamc@138 | 792 Ltac matcher := |

adamc@138 | 793 intros; |

adamc@204 | 794 repeat search_prem ltac:(simple apply False_prem || (simple apply ex_prem; intro)); |

adamc@204 | 795 repeat search_conc ltac:(simple apply True_conc || simple eapply ex_conc |

adamc@204 | 796 || search_prem ltac:(simple apply Match)); |

adamc@204 | 797 try simple apply imp_True. |

adamc@141 | 798 (* end thide *) |

adamc@140 | 799 |

adamc@140 | 800 (** Our tactic succeeds at proving a simple example. *) |

adamc@138 | 801 |

adamc@138 | 802 Theorem t2 : forall P Q : Prop, |

adamc@138 | 803 Q /\ (P /\ False) /\ P --> P /\ Q. |

adamc@138 | 804 matcher. |

adamc@138 | 805 Qed. |

adamc@138 | 806 |

adamc@140 | 807 (** In the generated proof, we find a trace of the workings of the search tactics. *) |

adamc@140 | 808 |

adamc@140 | 809 Print t2. |

adamc@220 | 810 (** %\vspace{-.15in}% [[ |

adamc@140 | 811 t2 = |

adamc@140 | 812 fun P Q : Prop => |

adamc@140 | 813 comm_prem (assoc_prem1 (assoc_prem2 (False_prem (P:=P /\ P /\ Q) (P /\ Q)))) |

adamc@140 | 814 : forall P Q : Prop, Q /\ (P /\ False) /\ P --> P /\ Q |

adamc@220 | 815 |

adamc@220 | 816 ]] |

adamc@140 | 817 |

adamc@220 | 818 We can also see that [matcher] is well-suited for cases where some human intervention is needed after the automation finishes. *) |

adamc@140 | 819 |

adamc@138 | 820 Theorem t3 : forall P Q R : Prop, |

adamc@138 | 821 P /\ Q --> Q /\ R /\ P. |

adamc@138 | 822 matcher. |

adamc@140 | 823 (** [[ |

adamc@140 | 824 ============================ |

adamc@140 | 825 True --> R |

adamc@220 | 826 |

adamc@140 | 827 ]] |

adamc@140 | 828 |

adamc@140 | 829 [matcher] canceled those conjuncts that it was able to cancel, leaving a simplified subgoal for us, much as [intuition] does. *) |

adamc@220 | 830 |

adamc@138 | 831 Abort. |

adamc@138 | 832 |

adamc@140 | 833 (** [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 | 834 |

adamc@138 | 835 Theorem t4 : forall (P : nat -> Prop) Q, (exists x, P x /\ Q) --> Q /\ (exists x, P x). |

adamc@138 | 836 matcher. |

adamc@138 | 837 Qed. |

adamc@138 | 838 |

adamc@140 | 839 Print t4. |

adamc@220 | 840 (** %\vspace{-.15in}% [[ |

adamc@140 | 841 t4 = |

adamc@140 | 842 fun (P : nat -> Prop) (Q : Prop) => |

adamc@140 | 843 and_True_prem |

adamc@140 | 844 (ex_prem (P:=fun x : nat => P x /\ Q) |

adamc@140 | 845 (fun x : nat => |

adamc@140 | 846 assoc_prem2 |

adamc@140 | 847 (Match (P:=Q) |

adamc@140 | 848 (and_True_conc |

adamc@140 | 849 (ex_conc (fun x0 : nat => P x0) x |

adamc@140 | 850 (Match (P:=P x) (imp_True (P:=True)))))))) |

adamc@140 | 851 : forall (P : nat -> Prop) (Q : Prop), |

adamc@140 | 852 (exists x : nat, P x /\ Q) --> Q /\ (exists x : nat, P x) |

adamc@140 | 853 ]] *) |

adamc@234 | 854 |

adamc@234 | 855 |

adamc@234 | 856 (** * Creating Unification Variables *) |

adamc@234 | 857 |

adamc@234 | 858 (** 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 | 859 |

adamc@234 | 860 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 | 861 |

adamc@234 | 862 Before we are ready to write a tactic, we can try out its ingredients one at a time. *) |

adamc@234 | 863 |

adamc@234 | 864 Theorem t5 : (forall x : nat, S x > x) -> 2 > 1. |

adamc@234 | 865 intros. |

adamc@234 | 866 |

adamc@234 | 867 (** [[ |

adamc@234 | 868 H : forall x : nat, S x > x |

adamc@234 | 869 ============================ |

adamc@234 | 870 2 > 1 |

adamc@234 | 871 |

adamc@234 | 872 ]] |

adamc@234 | 873 |

adamc@234 | 874 To instantiate [H] generically, we first need to name the value to be used for [x]. *) |

adamc@234 | 875 |

adamc@234 | 876 evar (y : nat). |

adamc@234 | 877 |

adamc@234 | 878 (** [[ |

adamc@234 | 879 H : forall x : nat, S x > x |

adamc@234 | 880 y := ?279 : nat |

adamc@234 | 881 ============================ |

adamc@234 | 882 2 > 1 |

adamc@234 | 883 |

adamc@234 | 884 ]] |

adamc@234 | 885 |

adamc@234 | 886 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 | 887 |

adamc@234 | 888 let y' := eval cbv delta [y] in y in |

adamc@234 | 889 clear y; generalize (H y'). |

adamc@234 | 890 |

adamc@234 | 891 (** [[ |

adamc@234 | 892 H : forall x : nat, S x > x |

adamc@234 | 893 ============================ |

adamc@234 | 894 S ?279 > ?279 -> 2 > 1 |

adamc@234 | 895 |

adamc@234 | 896 ]] |

adamc@234 | 897 |

adamc@234 | 898 Our instantiation was successful. We can finish by using the refined formula to replace the original. *) |

adamc@234 | 899 |

adamc@234 | 900 clear H; intro H. |

adamc@234 | 901 |

adamc@234 | 902 (** [[ |

adamc@234 | 903 H : S ?281 > ?281 |

adamc@234 | 904 ============================ |

adamc@234 | 905 2 > 1 |

adamc@234 | 906 |

adamc@234 | 907 ]] |

adamc@234 | 908 |

adamc@234 | 909 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 | 910 |

adamc@234 | 911 apply H. |

adamc@234 | 912 Qed. |

adamc@234 | 913 |

adamc@234 | 914 (** Now we can write a tactic that encapsulates the pattern we just employed, instantiating all quantifiers of a particular hypothesis. *) |

adamc@234 | 915 |

adamc@234 | 916 Ltac insterU H := |

adamc@234 | 917 repeat match type of H with |

adamc@234 | 918 | forall x : ?T, _ => |

adamc@234 | 919 let x := fresh "x" in |

adamc@234 | 920 evar (x : T); |

adamc@234 | 921 let x' := eval cbv delta [x] in x in |

adamc@234 | 922 clear x; generalize (H x'); clear H; intro H |

adamc@234 | 923 end. |

adamc@234 | 924 |

adamc@234 | 925 Theorem t5' : (forall x : nat, S x > x) -> 2 > 1. |

adamc@234 | 926 intro H; insterU H; apply H. |

adamc@234 | 927 Qed. |

adamc@234 | 928 |

adamc@234 | 929 (** 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 | 930 |

adamc@234 | 931 Ltac insterKeep H := |

adamc@234 | 932 let H' := fresh "H'" in |

adamc@234 | 933 generalize H; intro H'; insterU H'. |

adamc@234 | 934 |

adamc@234 | 935 Section t6. |

adamc@234 | 936 Variables A B : Type. |

adamc@234 | 937 Variable P : A -> B -> Prop. |

adamc@234 | 938 Variable f : A -> A -> A. |

adamc@234 | 939 Variable g : B -> B -> B. |

adamc@234 | 940 |

adamc@234 | 941 Hypothesis H1 : forall v, exists u, P v u. |

adamc@234 | 942 Hypothesis H2 : forall v1 u1 v2 u2, |

adamc@234 | 943 P v1 u1 |

adamc@234 | 944 -> P v2 u2 |

adamc@234 | 945 -> P (f v1 v2) (g u1 u2). |

adamc@234 | 946 |

adamc@234 | 947 Theorem t6 : forall v1 v2, exists u1, exists u2, P (f v1 v2) (g u1 u2). |

adamc@234 | 948 intros. |

adamc@234 | 949 |

adamc@234 | 950 (** 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 | 951 |

adamc@234 | 952 do 2 insterKeep H1. |

adamc@234 | 953 |

adamc@234 | 954 (** Our proof state is extended with two generic instances of [H1]. |

adamc@234 | 955 |

adamc@234 | 956 [[ |

adamc@234 | 957 H' : exists u : B, P ?4289 u |

adamc@234 | 958 H'0 : exists u : B, P ?4288 u |

adamc@234 | 959 ============================ |

adamc@234 | 960 exists u1 : B, exists u2 : B, P (f v1 v2) (g u1 u2) |

adamc@234 | 961 |

adamc@234 | 962 ]] |

adamc@234 | 963 |

adamc@234 | 964 [eauto] still cannot prove the goal, so we eliminate the two new existential quantifiers. *) |

adamc@234 | 965 |

adamc@234 | 966 repeat match goal with |

adamc@234 | 967 | [ H : ex _ |- _ ] => destruct H |

adamc@234 | 968 end. |

adamc@234 | 969 |

adamc@234 | 970 (** Now the goal is simple enough to solve by logic programming. *) |

adamc@234 | 971 |

adamc@234 | 972 eauto. |

adamc@234 | 973 Qed. |

adamc@234 | 974 End t6. |

adamc@234 | 975 |

adamc@234 | 976 (** 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 | 977 |

adamc@234 | 978 Section t7. |

adamc@234 | 979 Variables A B : Type. |

adamc@234 | 980 Variable Q : A -> Prop. |

adamc@234 | 981 Variable P : A -> B -> Prop. |

adamc@234 | 982 Variable f : A -> A -> A. |

adamc@234 | 983 Variable g : B -> B -> B. |

adamc@234 | 984 |

adamc@234 | 985 Hypothesis H1 : forall v, Q v -> exists u, P v u. |

adamc@234 | 986 Hypothesis H2 : forall v1 u1 v2 u2, |

adamc@234 | 987 P v1 u1 |

adamc@234 | 988 -> P v2 u2 |

adamc@234 | 989 -> P (f v1 v2) (g u1 u2). |

adamc@234 | 990 |

adamc@234 | 991 Theorem t6 : forall v1 v2, Q v1 -> Q v2 -> exists u1, exists u2, P (f v1 v2) (g u1 u2). |

adamc@234 | 992 intros; do 2 insterKeep H1; |

adamc@234 | 993 repeat match goal with |

adamc@234 | 994 | [ H : ex _ |- _ ] => destruct H |

adamc@234 | 995 end; eauto. |

adamc@234 | 996 |

adamc@234 | 997 (** This proof script does not hit any errors until the very end, when an error message like this one is displayed. |

adamc@234 | 998 |

adamc@234 | 999 [[ |

adamc@234 | 1000 No more subgoals but non-instantiated existential variables : |

adamc@234 | 1001 Existential 1 = |

adamc@234 | 1002 ?4384 : [A : Type |

adamc@234 | 1003 B : Type |

adamc@234 | 1004 Q : A -> Prop |

adamc@234 | 1005 P : A -> B -> Prop |

adamc@234 | 1006 f : A -> A -> A |

adamc@234 | 1007 g : B -> B -> B |

adamc@234 | 1008 H1 : forall v : A, Q v -> exists u : B, P v u |

adamc@234 | 1009 H2 : forall (v1 : A) (u1 : B) (v2 : A) (u2 : B), |

adamc@234 | 1010 P v1 u1 -> P v2 u2 -> P (f v1 v2) (g u1 u2) |

adamc@234 | 1011 v1 : A |

adamc@234 | 1012 v2 : A |

adamc@234 | 1013 H : Q v1 |

adamc@234 | 1014 H0 : Q v2 |

adamc@234 | 1015 H' : Q v2 -> exists u : B, P v2 u |- Q v2] |

adamc@234 | 1016 |

adamc@234 | 1017 ]] |

adamc@234 | 1018 |

adamc@234 | 1019 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 | 1020 |

adamc@234 | 1021 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 | 1022 |

adamc@234 | 1023 Abort. |

adamc@234 | 1024 End t7. |

adamc@234 | 1025 |

adamc@234 | 1026 Reset insterU. |

adamc@234 | 1027 |

adamc@234 | 1028 (** 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 | 1029 |

adamc@234 | 1030 Ltac insterU tac H := |

adamc@234 | 1031 repeat match type of H with |

adamc@234 | 1032 | forall x : ?T, _ => |

adamc@234 | 1033 match type of T with |

adamc@234 | 1034 | Prop => |

adamc@234 | 1035 (let H' := fresh "H'" in |

adamc@234 | 1036 assert (H' : T); [ |

adamc@234 | 1037 solve [ tac ] |

adamc@234 | 1038 | generalize (H H'); clear H H'; intro H ]) |

adamc@234 | 1039 || fail 1 |

adamc@234 | 1040 | _ => |

adamc@234 | 1041 let x := fresh "x" in |

adamc@234 | 1042 evar (x : T); |

adamc@234 | 1043 let x' := eval cbv delta [x] in x in |

adamc@234 | 1044 clear x; generalize (H x'); clear H; intro H |

adamc@234 | 1045 end |

adamc@234 | 1046 end. |

adamc@234 | 1047 |

adamc@234 | 1048 Ltac insterKeep tac H := |

adamc@234 | 1049 let H' := fresh "H'" in |

adamc@234 | 1050 generalize H; intro H'; insterU tac H'. |

adamc@234 | 1051 |

adamc@234 | 1052 Section t7. |

adamc@234 | 1053 Variables A B : Type. |

adamc@234 | 1054 Variable Q : A -> Prop. |

adamc@234 | 1055 Variable P : A -> B -> Prop. |

adamc@234 | 1056 Variable f : A -> A -> A. |

adamc@234 | 1057 Variable g : B -> B -> B. |

adamc@234 | 1058 |

adamc@234 | 1059 Hypothesis H1 : forall v, Q v -> exists u, P v u. |

adamc@234 | 1060 Hypothesis H2 : forall v1 u1 v2 u2, |

adamc@234 | 1061 P v1 u1 |

adamc@234 | 1062 -> P v2 u2 |

adamc@234 | 1063 -> P (f v1 v2) (g u1 u2). |

adamc@234 | 1064 |

adamc@234 | 1065 Theorem t6 : forall v1 v2, Q v1 -> Q v2 -> exists u1, exists u2, P (f v1 v2) (g u1 u2). |

adamc@234 | 1066 |

adamc@234 | 1067 (** 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 | 1068 |

adamc@234 | 1069 intros; do 2 insterKeep ltac:(idtac; match goal with |

adamc@234 | 1070 | [ H : Q ?v |- _ ] => |

adamc@234 | 1071 match goal with |

adamc@234 | 1072 | [ _ : context[P v _] |- _ ] => fail 1 |

adamc@234 | 1073 | _ => apply H |

adamc@234 | 1074 end |

adamc@234 | 1075 end) H1; |

adamc@234 | 1076 repeat match goal with |

adamc@234 | 1077 | [ H : ex _ |- _ ] => destruct H |

adamc@234 | 1078 end; eauto. |

adamc@234 | 1079 Qed. |

adamc@234 | 1080 End t7. |

adamc@234 | 1081 |

adamc@234 | 1082 (** It is often useful to instantiate existential variables explicitly. A built-in tactic provides one way of doing so. *) |

adamc@234 | 1083 |

adamc@234 | 1084 Theorem t8 : exists p : nat * nat, fst p = 3. |

adamc@234 | 1085 econstructor; instantiate (1 := (3, 2)); reflexivity. |

adamc@234 | 1086 Qed. |

adamc@234 | 1087 |

adamc@234 | 1088 (** 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 | 1089 |

adamc@234 | 1090 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 | 1091 |

adamc@234 | 1092 Ltac equate x y := |

adamc@234 | 1093 let H := fresh "H" in |

adamc@234 | 1094 assert (H : x = y); [ reflexivity | clear H ]. |

adamc@234 | 1095 |

adamc@234 | 1096 (** [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 | 1097 |

adamc@234 | 1098 Theorem t9 : exists p : nat * nat, fst p = 3. |

adamc@234 | 1099 econstructor; match goal with |

adamc@234 | 1100 | [ |- fst ?x = 3 ] => equate x (3, 2) |

adamc@234 | 1101 end; reflexivity. |

adamc@234 | 1102 Qed. |