### annotate src/Match.v @ 328:cbeccef45f4e

Pass over Match
author Adam Chlipala Sun, 25 Sep 2011 13:20:56 -0400 06d11a6363cd d7178fb77321
rev   line source
adamc@132 10 (* begin hide *)
adamc@132 16 (* end hide *)
adam@324 19 (** %\chapter{Proof Search in Ltac}% *)
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 23 (** * Some Built-In Automation Tactics *)
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.
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.
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 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@135 34 (** * Ltac Programming Basics *)
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 38 One common use for [match] tactics is identification of subjects for case analysis, as we see in this tactic definition. *)
adamc@141 40 (* begin thide *)
adamc@135 43 | [ |- if ?X then _ else _ ] => destruct X
adamc@141 45 (* end thide *)
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 49 Theorem hmm : forall (a b c : bool),
adamc@141 57 (* begin thide *)
adamc@135 58 intros; repeat find_if; constructor.
adamc@141 60 (* end thide *)
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.
adam@328 64 Another very useful Ltac building block is %\index{context patterns}\textit{%#<i>#context patterns#</i>#%}%. *)
adamc@141 66 (* begin thide *)
adamc@135 69 | [ |- context[if ?X then _ else _] ] => destruct X
adamc@141 71 (* end thide *)
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 75 Theorem hmm' : forall (a b c : bool),
adamc@141 83 (* begin thide *)
adamc@135 84 intros; repeat find_if_inside; constructor.
adamc@141 86 (* end thide *)
adamc@135 88 (** We can also use [find_if_inside] to prove goals that [find_if] does not simplify sufficiently. *)
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@141 95 (* end thide *)
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@141 99 (* begin thide *)
adamc@135 101 repeat match goal with
adamc@135 102 | [ H : ?P |- ?P ] => exact H
adamc@135 104 | [ |- True ] => constructor
adamc@135 105 | [ |- _ /\ _ ] => constructor
adamc@135 106 | [ |- _ -> _ ] => intro
adamc@135 108 | [ H : False |- _ ] => destruct H
adamc@135 109 | [ H : _ /\ _ |- _ ] => destruct H
adamc@135 110 | [ H : _ \/ _ |- _ ] => destruct H
adam@328 112 | [ H1 : ?P -> ?Q, H2 : ?P |- _ ] => specialize (H1 H2)
adamc@141 114 (* end thide *)
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.
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].
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@135 123 Variables P Q R : Prop.
adamc@138 125 Theorem propositional : (P \/ Q \/ False) /\ (P -> Q) -> True /\ Q.
adamc@141 126 (* begin thide *)
adamc@141 129 (* end thide *)
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@135 134 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.
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@135 138 For instance, this (unnecessarily verbose) proof script works: *)
adamc@135 140 Theorem m1 : True.
adamc@135 142 | [ |- _ ] => intro
adamc@135 143 | [ |- True ] => constructor
adamc@141 145 (* begin thide *)
adamc@141 147 (* end thide *)
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@135 151 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@135 153 Theorem m2 : forall P Q R : Prop, P -> Q -> R -> Q.
adamc@135 154 intros; match goal with
adamc@220 155 | [ H : _ |- _ ] => idtac H
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@141 160 (* begin thide *)
adamc@135 162 | [ H : _ |- _ ] => exact H
adamc@141 165 (* end thide *)
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@135 169 (** Now we are equipped to implement a tactic for checking that a proposition is not among our hypotheses: *)
adamc@141 171 (* begin thide *)
adamc@135 172 Ltac notHyp P :=
adamc@135 174 | [ _ : P |- _ ] => fail 1
adamc@135 177 | ?P1 /\ ?P2 => first [ notHyp P1 | notHyp P2 | fail 2 ]
adamc@135 178 | _ => idtac
adamc@141 181 (* end thide *)
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.
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.
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@135 189 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@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 *)
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@135 199 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@141 201 (* begin thide *)
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
adam@328 209 | [ H : forall x, ?P x -> _, H' : ?P ?X |- _ ] => extend (H X H')
adamc@141 211 (* end thide *)
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@135 215 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@135 217 We can check that [completer] is working properly: *)
adamc@135 220 Variable A : Set.
adamc@135 221 Variables P Q R S : A -> Prop.
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@135 226 Theorem fo : forall x, P x -> S x.
adamc@141 227 (* begin thide *)
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@141 242 (* end thide *)
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@141 247 (* begin thide *)
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
adam@328 255 | [ H : forall x, ?P x -> _, H' : ?P ?X |- _ ] => extend (H X H')
adamc@141 257 (* end thide *)
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@135 262 Variable A : Set.
adamc@135 263 Variables P Q R S : A -> Prop.
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@135 268 Theorem fo' : forall x, P x -> S x.
adamc@141 269 (* begin thide *)
adamc@135 275 Coq loops forever at this point. What went wrong? *)
adamc@141 278 (* end thide *)
adamc@136 281 (** A few examples should illustrate the issue. Here we see a [match]-based proof that works fine: *)
adamc@136 283 Theorem t1 : forall x : nat, x = x.
adamc@136 285 | [ |- forall x, _ ] => trivial
adamc@141 287 (* begin thide *)
adamc@141 289 (* end thide *)
adamc@136 291 (** This one fails. *)
adamc@141 293 (* begin thide *)
adamc@136 294 Theorem t1' : forall x : nat, x = x.
adamc@136 297 | [ |- forall x, ?P ] => trivial
adamc@136 302 User error: No matching clauses for match goal
adamc@141 307 (* end thide *)
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.
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.
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@137 316 (** * Functional Programming in Ltac *)
adamc@141 318 (* EX: Write a list length function in Ltac. *)
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@137 322 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@137 325 Ltac length ls :=
adamc@137 327 | nil => O
adamc@137 328 | _ :: ls' => S (length ls')
adamc@137 336 At this point, we hopefully remember that pattern variable names must be prefixed by question marks in Ltac.
adamc@137 339 Ltac length ls :=
adamc@137 341 | nil => O
adamc@137 342 | _ :: ?ls' => S (length ls')
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@141 352 (* begin thide *)
adamc@137 353 Ltac length ls :=
adamc@137 355 | nil => O
adamc@137 356 | _ :: ?ls' => constr:(S (length ls'))
adamc@137 359 (** This definition is accepted. It can be a little awkward to test Ltac definitions like this. Here is one method. *)
adamc@137 362 let n := length (1 :: 2 :: 3 :: nil) in
adamc@137 365 n := S (length (2 :: 3 :: nil)) : nat
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.
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. *)
adam@328 377 (* begin hide *)
adam@328 379 (* end hide *)
adam@328 380 (** %\noindent\coqdockw{%#<tt>#Reset#</tt>#%}% [length.] *)
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@137 384 Ltac length ls :=
adamc@137 386 | nil => O
adamc@137 387 | _ :: ?ls' =>
adamc@137 388 let ls'' := length ls' in
adamc@137 393 let n := length (1 :: 2 :: 3 :: nil) in
adamc@137 396 n := 3 : nat
adamc@141 403 (* end thide *)
adamc@141 405 (* EX: Write a list map function in Ltac. *)
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@141 409 (* begin thide *)
adamc@137 410 Ltac map T f :=
adamc@137 411 let rec map' ls :=
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'')
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'')].
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]. *)
adam@288 426 let ls := map (nat * nat)%type ltac:(fun x => constr:( x, x)) (1 :: 2 :: 3 :: nil) in
adamc@137 429 l := (1, 1) :: (2, 2) :: (3, 3) :: nil : list (nat * nat)
adamc@141 436 (* end thide *)
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@328 440 (* begin hide *)
adam@328 442 (* end hide *)
adam@328 443 (** %\noindent\coqdockw{%#<tt>#Reset#</tt>#%}% [length.] *)
adam@328 445 Ltac length ls :=
adam@328 448 | nil => O
adam@328 449 | _ :: ?ls' =>
adam@328 450 let ls'' := length ls' in
adam@328 454 (** Coq accepts the tactic definition, but the code is fatally flawed and will always lead to dynamic type errors. *)
adam@328 458 let n := length (1 :: 2 :: 3 :: nil) in
adam@328 463 Error: variable n should be bound to a term.
adam@328 467 (** 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{monads}\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 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 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 (* begin hide *)
adam@328 475 (* end hide *)
adam@328 476 (** %\noindent\coqdockw{%#<tt>#Reset#</tt>#%}% [length.] *)
adam@328 478 Ltac length ls k :=
adam@328 481 | nil => k O
adam@328 482 | _ :: ?ls' => length ls' ltac:(fun n => k (S n))
adam@328 485 (** 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 length (1 :: 2 :: 3 :: nil) ltac:(fun n => pose n).
adam@328 490 (1 :: 2 :: 3 :: nil)
adam@328 491 (2 :: 3 :: nil)
adam@328 498 (** 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]. *)
adamc@139 501 (** * Recursive Proof Search *)
adamc@139 503 (** 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.
adam@288 505 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@141 507 (* begin thide *)
adamc@139 508 Ltac inster n :=
adamc@139 511 | S ?n' =>
adamc@139 513 | [ H : forall x : ?T, _, x : ?T |- _ ] => generalize (H x); inster n'
adamc@141 516 (* end thide *)
adam@328 518 (** 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@139 520 We can verify the efficacy of [inster] with two short examples. The built-in [firstorder] tactic (with no extra arguments) is able to prove the first but not the second. *)
adamc@139 523 Variable A : Set.
adamc@139 524 Variables P Q : A -> Prop.
adamc@139 525 Variable f : A -> A.
adamc@139 526 Variable g : A -> A -> A.
adamc@139 528 Hypothesis H1 : forall x y, P (g x y) -> Q (f x).
adam@328 530 Theorem test_inster : forall x, P (g x x) -> Q (f x).
adamc@139 534 Hypothesis H3 : forall u v, P u /\ P v /\ u <> v -> P (g u v).
adamc@139 535 Hypothesis H4 : forall u, Q (f u) -> P u /\ P (f u).
adamc@139 537 Theorem test_inster2 : forall x y, x <> y -> P x -> Q (f y) -> Q (f x).
adam@328 542 (** 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.
adam@288 544 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.
adam@328 546 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.
adam@288 548 To distinguish the implications that our tactic handles from the implications that will show up as %%#"#plumbing#"#%''% in various lemmas, we define a wrapper definition, a notation, and a tactic. *)
adamc@138 550 Definition imp (P1 P2 : Prop) := P1 -> P2.
adamc@140 551 Infix "-->" := imp (no associativity, at level 95).
adamc@140 552 Ltac imp := unfold imp; firstorder.
adamc@140 554 (** These lemmas about [imp] will be useful in the tactic that we will write. *)
adamc@138 556 Theorem and_True_prem : forall P Q,
adamc@138 557 (P /\ True --> Q)
adamc@138 558 -> (P --> Q).
adamc@138 562 Theorem and_True_conc : forall P Q,
adamc@138 563 (P --> Q /\ True)
adamc@138 564 -> (P --> Q).
adamc@138 568 Theorem assoc_prem1 : forall P Q R S,
adamc@138 569 (P /\ (Q /\ R) --> S)
adamc@138 570 -> ((P /\ Q) /\ R --> S).
adamc@138 574 Theorem assoc_prem2 : forall P Q R S,
adamc@138 575 (Q /\ (P /\ R) --> S)
adamc@138 576 -> ((P /\ Q) /\ R --> S).
adamc@138 580 Theorem comm_prem : forall P Q R,
adamc@138 581 (P /\ Q --> R)
adamc@138 582 -> (Q /\ P --> R).
adamc@138 586 Theorem assoc_conc1 : forall P Q R S,
adamc@138 587 (S --> P /\ (Q /\ R))
adamc@138 588 -> (S --> (P /\ Q) /\ R).
adamc@138 592 Theorem assoc_conc2 : forall P Q R S,
adamc@138 593 (S --> Q /\ (P /\ R))
adamc@138 594 -> (S --> (P /\ Q) /\ R).
adamc@138 598 Theorem comm_conc : forall P Q R,
adamc@138 599 (R --> P /\ Q)
adamc@138 600 -> (R --> Q /\ P).
adam@288 604 (** 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@138 606 Ltac search_prem tac :=
adamc@138 607 let rec search P :=
adamc@138 609 || (apply and_True_prem; tac)
adamc@138 610 || match P with
adamc@138 611 | ?P1 /\ ?P2 =>
adamc@138 612 (apply assoc_prem1; search P1)
adamc@138 613 || (apply assoc_prem2; search P2)
adamc@138 615 in match goal with
adamc@138 616 | [ |- ?P /\ _ --> _ ] => search P
adamc@138 617 | [ |- _ /\ ?P --> _ ] => apply comm_prem; search P
adamc@138 618 | [ |- _ --> _ ] => progress (tac || (apply and_True_prem; tac))
adam@328 621 (** 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.
adam@328 623 The [search] function itself tries the same tricks as in the last case of the final [match]. Additionally, if neither works, it checks if [P] is a conjunction. If so, it calls itself recursively on each conjunct, first applying associativity lemmas to maintain the goal-form invariant.
adamc@140 625 We will also want a dual function [search_conc], which does tree search through an [imp] conclusion. *)
adamc@138 627 Ltac search_conc tac :=
adamc@138 628 let rec search P :=
adamc@138 630 || (apply and_True_conc; tac)
adamc@138 631 || match P with
adamc@138 632 | ?P1 /\ ?P2 =>
adamc@138 633 (apply assoc_conc1; search P1)
adamc@138 634 || (apply assoc_conc2; search P2)
adamc@138 636 in match goal with
adamc@138 637 | [ |- _ --> ?P /\ _ ] => search P
adamc@138 638 | [ |- _ --> _ /\ ?P ] => apply comm_conc; search P
adamc@138 639 | [ |- _ --> _ ] => progress (tac || (apply and_True_conc; tac))
adamc@140 642 (** 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]. *)
adam@328 644 (* begin thide *)
adamc@138 645 Theorem False_prem : forall P Q,
adamc@138 646 False /\ P --> Q.
adamc@138 650 Theorem True_conc : forall P Q : Prop,
adamc@138 652 -> (P --> True /\ Q).
adamc@138 656 Theorem Match : forall P Q R : Prop,
adamc@138 658 -> (P /\ Q --> P /\ R).
adamc@138 662 Theorem ex_prem : forall (T : Type) (P : T -> Prop) (Q R : Prop),
adamc@138 663 (forall x, P x /\ Q --> R)
adamc@138 664 -> (ex P /\ Q --> R).
adamc@138 668 Theorem ex_conc : forall (T : Type) (P : T -> Prop) (Q R : Prop) x,
adamc@138 669 (Q --> P x /\ R)
adamc@138 670 -> (Q --> ex P /\ R).
adam@288 674 (** We will also want a %%#"#base case#"#%''% lemma for finishing proofs where cancelation has removed every constituent of the conclusion. *)
adamc@138 676 Theorem imp_True : forall P,
adamc@220 681 (** 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. *)
adam@288 685 repeat search_prem ltac:( simple apply False_prem || ( simple apply ex_prem; intro));
adam@288 686 repeat search_conc ltac:( simple apply True_conc || simple eapply ex_conc
adam@288 687 || search_prem ltac:( simple apply Match));
adamc@204 688 try simple apply imp_True.
adamc@141 689 (* end thide *)
adamc@140 691 (** Our tactic succeeds at proving a simple example. *)
adamc@138 693 Theorem t2 : forall P Q : Prop,
adamc@138 694 Q /\ (P /\ False) /\ P --> P /\ Q.
adamc@140 698 (** In the generated proof, we find a trace of the workings of the search tactics. *)
adamc@140 703 fun P Q : Prop =>
adamc@140 704 comm_prem (assoc_prem1 (assoc_prem2 (False_prem (P:=P /\ P /\ Q) (P /\ Q))))
adamc@140 705 : forall P Q : Prop, Q /\ (P /\ False) /\ P --> P /\ Q
adamc@220 709 We can also see that [matcher] is well-suited for cases where some human intervention is needed after the automation finishes. *)
adamc@138 711 Theorem t3 : forall P Q R : Prop,
adamc@138 712 P /\ Q --> Q /\ R /\ P.
adam@328 720 Our tactic canceled those conjuncts that it was able to cancel, leaving a simplified subgoal for us, much as [intuition] does. *)
adam@328 724 (** 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@138 726 Theorem t4 : forall (P : nat -> Prop) Q, (exists x, P x /\ Q) --> Q /\ (exists x, P x).
adamc@140 733 fun (P : nat -> Prop) (Q : Prop) =>
adamc@140 735 (ex_prem (P:=fun x : nat => P x /\ Q)
adamc@140 736 (fun x : nat =>
adamc@140 740 (ex_conc (fun x0 : nat => P x0) x
adamc@140 741 (Match (P:=P x) (imp_True (P:=True))))))))
adamc@140 742 : forall (P : nat -> Prop) (Q : Prop),
adamc@140 743 (exists x : nat, P x /\ Q) --> Q /\ (exists x : nat, P x)
adamc@234 748 (** * Creating Unification Variables *)
adamc@234 750 (** 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.
adam@328 752 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@234 754 Before we are ready to write a tactic, we can try out its ingredients one at a time. *)
adamc@234 756 Theorem t5 : (forall x : nat, S x > x) -> 2 > 1.
adamc@234 760 H : forall x : nat, S x > x
adam@328 766 To instantiate [H] generically, we first need to name the value to be used for [x].%\index{tactics!evar}% *)
adamc@234 768 evar (y : nat).
adamc@234 771 H : forall x : nat, S x > x
adamc@234 772 y := ?279 : nat
adam@328 778 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.). *)
adam@328 780 let y' := eval unfold y in y in
adamc@234 781 clear y; generalize (H y').
adamc@234 784 H : forall x : nat, S x > x
adamc@234 786 S ?279 > ?279 -> 2 > 1
adamc@234 790 Our instantiation was successful. We can finish by using the refined formula to replace the original. *)
adamc@234 792 clear H; intro H.
adamc@234 795 H : S ?281 > ?281
adamc@234 801 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 (** Now we can write a tactic that encapsulates the pattern we just employed, instantiating all quantifiers of a particular hypothesis. *)
adamc@234 808 Ltac insterU H :=
adamc@234 809 repeat match type of H with
adamc@234 810 | forall x : ?T, _ =>
adamc@234 811 let x := fresh "x" in
adamc@234 812 evar (x : T);
adam@328 813 let x' := eval unfold x in x in
adam@328 814 clear x; specialize (H x')
adamc@234 817 Theorem t5' : (forall x : nat, S x > x) -> 2 > 1.
adamc@234 818 intro H; insterU H; apply H.
adam@328 821 (** 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@234 823 Ltac insterKeep H :=
adamc@234 824 let H' := fresh "H'" in
adamc@234 825 generalize H; intro H'; insterU H'.
adamc@234 828 Variables A B : Type.
adamc@234 829 Variable P : A -> B -> Prop.
adamc@234 830 Variable f : A -> A -> A.
adamc@234 831 Variable g : B -> B -> B.
adamc@234 833 Hypothesis H1 : forall v, exists u, P v u.
adamc@234 834 Hypothesis H2 : forall v1 u1 v2 u2,
adamc@234 836 -> P v2 u2
adamc@234 837 -> P (f v1 v2) (g u1 u2).
adamc@234 839 Theorem t6 : forall v1 v2, exists u1, exists u2, P (f v1 v2) (g u1 u2).
adam@328 842 (** 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@234 844 do 2 insterKeep H1.
adamc@234 846 (** Our proof state is extended with two generic instances of [H1].
adamc@234 849 H' : exists u : B, P ?4289 u
adamc@234 850 H'0 : exists u : B, P ?4288 u
adamc@234 852 exists u1 : B, exists u2 : B, P (f v1 v2) (g u1 u2)
adam@328 856 [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@234 858 repeat match goal with
adamc@234 859 | [ H : ex _ |- _ ] => destruct H
adamc@234 862 (** Now the goal is simple enough to solve by logic programming. *)
adamc@234 868 (** 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@234 871 Variables A B : Type.
adamc@234 872 Variable Q : A -> Prop.
adamc@234 873 Variable P : A -> B -> Prop.
adamc@234 874 Variable f : A -> A -> A.
adamc@234 875 Variable g : B -> B -> B.
adamc@234 877 Hypothesis H1 : forall v, Q v -> exists u, P v u.
adamc@234 878 Hypothesis H2 : forall v1 u1 v2 u2,
adamc@234 880 -> P v2 u2
adamc@234 881 -> P (f v1 v2) (g u1 u2).
adam@297 883 Theorem t7 : forall v1 v2, Q v1 -> Q v2 -> exists u1, exists u2, P (f v1 v2) (g u1 u2).
adamc@234 884 intros; do 2 insterKeep H1;
adamc@234 885 repeat match goal with
adamc@234 886 | [ H : ex _ |- _ ] => destruct H
adamc@234 889 (** This proof script does not hit any errors until the very end, when an error message like this one is displayed.
adamc@234 892 No more subgoals but non-instantiated existential variables :
adamc@234 896 ?4384 : [A : Type
adamc@234 898 Q : A -> Prop
adamc@234 899 P : A -> B -> Prop
adamc@234 900 f : A -> A -> A
adamc@234 901 g : B -> B -> B
adamc@234 902 H1 : forall v : A, Q v -> exists u : B, P v u
adamc@234 903 H2 : forall (v1 : A) (u1 : B) (v2 : A) (u2 : B),
adamc@234 904 P v1 u1 -> P v2 u2 -> P (f v1 v2) (g u1 u2)
adamc@234 907 H : Q v1
adamc@234 908 H0 : Q v2
adamc@234 909 H' : Q v2 -> exists u : B, P v2 u |- Q v2]
adam@288 913 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@234 915 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]. *)
adam@328 920 (* begin hide *)
adam@328 922 (* end hide *)
adam@328 923 (** %\noindent\coqdockw{%#<tt>#Reset#</tt>#%}% [insterU.] *)
adam@328 925 (** 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@234 927 Ltac insterU tac H :=
adamc@234 928 repeat match type of H with
adamc@234 929 | forall x : ?T, _ =>
adamc@234 930 match type of T with
adamc@234 932 (let H' := fresh "H'" in
adam@328 933 assert (H' : T) by solve [ tac ];
adam@328 934 specialize (H H'); clear H')
adamc@234 937 let x := fresh "x" in
adamc@234 938 evar (x : T);
adam@328 939 let x' := eval unfold x in x in
adam@328 940 clear x; specialize (H x')
adamc@234 944 Ltac insterKeep tac H :=
adamc@234 945 let H' := fresh "H'" in
adamc@234 946 generalize H; intro H'; insterU tac H'.
adamc@234 949 Variables A B : Type.
adamc@234 950 Variable Q : A -> Prop.
adamc@234 951 Variable P : A -> B -> Prop.
adamc@234 952 Variable f : A -> A -> A.
adamc@234 953 Variable g : B -> B -> B.
adamc@234 955 Hypothesis H1 : forall v, Q v -> exists u, P v u.
adamc@234 956 Hypothesis H2 : forall v1 u1 v2 u2,
adamc@234 958 -> P v2 u2
adamc@234 959 -> P (f v1 v2) (g u1 u2).
adamc@234 961 Theorem t6 : forall v1 v2, Q v1 -> Q v2 -> exists u1, exists u2, P (f v1 v2) (g u1 u2).
adamc@234 963 (** 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@234 965 intros; do 2 insterKeep ltac:(idtac; match goal with
adamc@234 966 | [ H : Q ?v |- _ ] =>
adamc@234 968 | [ _ : context[P v _] |- _ ] => fail 1
adamc@234 969 | _ => apply H
adamc@234 972 repeat match goal with
adamc@234 973 | [ H : ex _ |- _ ] => destruct H
adamc@234 978 (** It is often useful to instantiate existential variables explicitly. A built-in tactic provides one way of doing so. *)
adamc@234 980 Theorem t8 : exists p : nat * nat, fst p = 3.
adamc@234 981 econstructor; instantiate (1 := (3, 2)); reflexivity.
adamc@234 984 (** 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 [:=].