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Co-equality
author Adam Chlipala <adamc@hcoop.net>
date Tue, 30 Sep 2008 17:07:57 -0400
parents 437cc4857e2a
children 739c2818d6e2
rev   line source
adamc@62 1 (* Copyright (c) 2008, Adam Chlipala
adamc@62 2 *
adamc@62 3 * This work is licensed under a
adamc@62 4 * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
adamc@62 5 * Unported License.
adamc@62 6 * The license text is available at:
adamc@62 7 * http://creativecommons.org/licenses/by-nc-nd/3.0/
adamc@62 8 *)
adamc@62 9
adamc@62 10 (* begin hide *)
adamc@62 11 Require Import List.
adamc@62 12
adamc@62 13 Require Import Tactics.
adamc@62 14
adamc@62 15 Set Implicit Arguments.
adamc@62 16 (* end hide *)
adamc@62 17
adamc@62 18
adamc@62 19 (** %\chapter{Infinite Data and Proofs}% *)
adamc@62 20
adamc@62 21 (** In lazy functional programming languages like Haskell, infinite data structures are everywhere. Infinite lists and more exotic datatypes provide convenient abstractions for communication between parts of a program. Achieving similar convenience without infinite lazy structures would, in many cases, require acrobatic inversions of control flow.
adamc@62 22
adamc@62 23 Laziness is easy to implement in Haskell, where all the definitions in a program may be thought of as mutually recursive. In such an unconstrained setting, it is easy to implement an infinite loop when you really meant to build an infinite list, where any finite prefix of the list should be forceable in finite time. Haskell programmers learn how to avoid such slip-ups. In Coq, such a laissez-faire policy is not good enough.
adamc@62 24
adamc@62 25 We spent some time in the last chapter discussing the Curry-Howard isomorphism, where proofs are identified with functional programs. In such a setting, infinite loops, intended or otherwise, are disastrous. If Coq allowed the full breadth of definitions that Haskell did, we could code up an infinite loop and use it to prove any proposition vacuously. That is, the addition of general recursion would make CIC %\textit{%#<i>#inconsistent#</i>#%}%.
adamc@62 26
adamc@62 27 There are also algorithmic considerations that make universal termination very desirable. We have seen how tactics like [reflexivity] compare terms up to equivalence under computational rules. Calls to recursive, pattern-matching functions are simplified automatically, with no need for explicit proof steps. It would be very hard to hold onto that kind of benefit if it became possible to write non-terminating programs; we would be running smack into the halting problem.
adamc@62 28
adamc@62 29 One solution is to use types to contain the possibility of non-termination. For instance, we can create a "non-termination monad," inside which we must write all of our general-recursive programs. In later chapters, we will spend some time on this idea and its applications. For now, we will just say that it is a heavyweight solution, and so we would like to avoid it whenever possible.
adamc@62 30
adamc@62 31 Luckily, Coq has special support for a class of lazy data structures that happens to contain most examples found in Haskell. That mechanism, %\textit{%#<i>#co-inductive types#</i>#%}%, is the subject of this chapter. *)
adamc@62 32
adamc@62 33
adamc@62 34 (** * Computing with Infinite Data *)
adamc@62 35
adamc@62 36 (** Let us begin with the most basic type of infinite data, %\textit{%#<i>#streams#</i>#%}%, or lazy lists. *)
adamc@62 37
adamc@62 38 Section stream.
adamc@62 39 Variable A : Set.
adamc@62 40
adamc@62 41 CoInductive stream : Set :=
adamc@62 42 | Cons : A -> stream -> stream.
adamc@62 43 End stream.
adamc@62 44
adamc@62 45 (** The definition is surprisingly simple. Starting from the definition of [list], we just need to change the keyword [Inductive] to [CoInductive]. We could have left a [Nil] constructor in our definition, but we will leave it out to force all of our streams to be infinite.
adamc@62 46
adamc@62 47 How do we write down a stream constant? Obviously simple application of constructors is not good enough, since we could only denote finite objects that way. Rather, whereas recursive definitions were necessary to %\textit{%#<i>#use#</i>#%}% values of recursive inductive types effectively, here we find that we need %\textit{%#<i>#co-recursive definitions#</i>#%}% to %\textit{%#<i>#build#</i>#%}% values of co-inductive types effectively.
adamc@62 48
adamc@62 49 We can define a stream consisting only of zeroes. *)
adamc@62 50
adamc@62 51 CoFixpoint zeroes : stream nat := Cons 0 zeroes.
adamc@62 52
adamc@62 53 (** We can also define a stream that alternates between [true] and [false]. *)
adamc@62 54
adamc@62 55 CoFixpoint trues : stream bool := Cons true falses
adamc@62 56 with falses : stream bool := Cons false trues.
adamc@62 57
adamc@62 58 (** Co-inductive values are fair game as arguments to recursive functions, and we can use that fact to write a function to take a finite approximation of a stream. *)
adamc@62 59
adamc@62 60 Fixpoint approx A (s : stream A) (n : nat) {struct n} : list A :=
adamc@62 61 match n with
adamc@62 62 | O => nil
adamc@62 63 | S n' =>
adamc@62 64 match s with
adamc@62 65 | Cons h t => h :: approx t n'
adamc@62 66 end
adamc@62 67 end.
adamc@62 68
adamc@62 69 Eval simpl in approx zeroes 10.
adamc@62 70 (** [[
adamc@62 71
adamc@62 72 = 0 :: 0 :: 0 :: 0 :: 0 :: 0 :: 0 :: 0 :: 0 :: 0 :: nil
adamc@62 73 : list nat
adamc@62 74 ]] *)
adamc@62 75 Eval simpl in approx trues 10.
adamc@62 76 (** [[
adamc@62 77
adamc@62 78 = true
adamc@62 79 :: false
adamc@62 80 :: true
adamc@62 81 :: false
adamc@62 82 :: true :: false :: true :: false :: true :: false :: nil
adamc@62 83 : list bool
adamc@62 84 ]] *)
adamc@62 85
adamc@62 86 (** So far, it looks like co-inductive types might be a magic bullet, allowing us to import all of the Haskeller's usual tricks. However, there are important restrictions that are dual to the restrictions on the use of inductive types. Fixpoints %\textit{%#<i>#consume#</i>#%}% values of inductive types, with restrictions on which %\textit{%#<i>#arguments#</i>#%}% may be passed in recursive calls. Dually, co-fixpoints %\textit{%#<i>#produce#</i>#%}% values of co-inductive types, with restrictions on what may be done with the %\textit{%#<i>#results#</i>#%}% of co-recursive calls.
adamc@62 87
adamc@62 88 The restriction for co-inductive types shows up as the %\textit{%#<i>#guardedness condition#</i>#%}%, and it can be broken into two parts. First, consider this stream definition, which would be legal in Haskell.
adamc@62 89
adamc@62 90 [[
adamc@62 91 CoFixpoint looper : stream nat := looper.
adamc@62 92 [[
adamc@62 93 Error:
adamc@62 94 Recursive definition of looper is ill-formed.
adamc@62 95 In environment
adamc@62 96 looper : stream nat
adamc@62 97
adamc@62 98 unguarded recursive call in "looper"
adamc@62 99 *)
adamc@62 100
adamc@62 101 (** The rule we have run afoul of here is that %\textit{%#<i>#every co-recursive call must be guarded by a constructor#</i>#%}%; that is, every co-recursive call must be a direct argument to a constructor of the co-inductive type we are generating. It is a good thing that this rule is enforced. If the definition of [looper] were accepted, our [approx] function would run forever when passed [looper], and we would have fallen into inconsistency.
adamc@62 102
adamc@62 103 The second rule of guardedness is easiest to see by first introducing a more complicated, but legal, co-fixpoint. *)
adamc@62 104
adamc@62 105 Section map.
adamc@62 106 Variables A B : Set.
adamc@62 107 Variable f : A -> B.
adamc@62 108
adamc@62 109 CoFixpoint map (s : stream A) : stream B :=
adamc@62 110 match s with
adamc@62 111 | Cons h t => Cons (f h) (map t)
adamc@62 112 end.
adamc@62 113 End map.
adamc@62 114
adamc@62 115 (** This code is a literal copy of that for the list [map] function, with the [Nil] case removed and [Fixpoint] changed to [CoFixpoint]. Many other standard functions on lazy data structures can be implemented just as easily. Some, like [filter], cannot be implemented. Since the predicate passed to [filter] may reject every element of the stream, we cannot satisfy even the first guardedness condition.
adamc@62 116
adamc@62 117 The second condition is subtler. To illustrate it, we start off with another co-recursive function definition that %\textit{%#<i>#is#</i>#%}% legal. The function [interleaves] takes two streams and produces a new stream that alternates between their elements. *)
adamc@62 118
adamc@62 119 Section interleave.
adamc@62 120 Variable A : Set.
adamc@62 121
adamc@62 122 CoFixpoint interleave (s1 s2 : stream A) : stream A :=
adamc@62 123 match s1, s2 with
adamc@62 124 | Cons h1 t1, Cons h2 t2 => Cons h1 (Cons h2 (interleave t1 t2))
adamc@62 125 end.
adamc@62 126 End interleave.
adamc@62 127
adamc@62 128 (** Now say we want to write a weird stuttering version of [map] that repeats elements in a particular way, based on interleaving. *)
adamc@62 129
adamc@62 130 Section map'.
adamc@62 131 Variables A B : Set.
adamc@62 132 Variable f : A -> B.
adamc@62 133
adamc@62 134 (** [[
adamc@62 135
adamc@62 136 CoFixpoint map' (s : stream A) : stream B :=
adamc@62 137 match s with
adamc@62 138 | Cons h t => interleave (Cons (f h) (map' s)) (Cons (f h) (map' s))
adamc@62 139 end. *)
adamc@62 140
adamc@62 141 (** We get another error message about an unguarded recursive call. This is because we are violating the second guardedness condition, which says that, not only must co-recursive calls be arguments to constructors, there must also %\textit{%#<i>#not be anything but [match]es and calls to constructors of the same co-inductive type#</i>#%}% wrapped around these immediate uses of co-recursive calls. The actual implemented rule for guardedness is a little more lenient than what we have just stated, but you can count on the illegality of any exception that would enhance the expressive power of co-recursion.
adamc@62 142
adamc@62 143 Why enforce a rule like this? Imagine that, instead of [interleave], we had called some other, less well-behaved function on streams. Perhaps this other function might be defined mutually with [map']. It might deconstruct its first argument, retrieving [map' s] from within [Cons (f h) (map' s)]. Next it might try a [match] on this retrieved value, which amounts to deconstructing [map' s]. To figure out how this [match] turns out, we need to know the top-level structure of [map' s], but this is exactly what we started out trying to determine! We run into a loop in the evaluation process, and we have reached a witness of inconsistency if we are evaluating [approx (map' s) 1] for any [s]. *)
adamc@62 144 End map'.
adamc@62 145
adamc@63 146
adamc@63 147 (** * Infinite Proofs *)
adamc@63 148
adamc@63 149 (** Let us say we want to give two different definitions of a stream of all ones, and then we want to prove that they are equivalent. *)
adamc@63 150
adamc@63 151 CoFixpoint ones : stream nat := Cons 1 ones.
adamc@63 152 Definition ones' := map S zeroes.
adamc@63 153
adamc@63 154 (** The obvious statement of the equality is this: *)
adamc@63 155
adamc@63 156 Theorem ones_eq : ones = ones'.
adamc@63 157
adamc@63 158 (** However, faced with the initial subgoal, it is not at all clear how this theorem can be proved. In fact, it is unprovable. The [eq] predicate that we use is fundamentally limited to equalities that can be demonstrated by finite, syntactic arguments. To prove this equivalence, we will need to introduce a new relation. *)
adamc@63 159 Abort.
adamc@63 160
adamc@63 161 (** Co-inductive datatypes make sense by analogy from Haskell. What we need now is a %\textit{%#<i>#co-inductive proposition#</i>#%}%. That is, we want to define a proposition whose proofs may be infinite, subject to the guardedness condition. The idea of infinite proofs does not show up in usual mathematics, but it can be very useful (unsurprisingly) for reasoning about infinite data structures. Besides examples from Haskell, infinite data and proofs will also turn out to be useful for modelling inherently infinite mathematical objects, like program executions.
adamc@63 162
adamc@63 163 We are ready for our first co-inductive predicate. *)
adamc@63 164
adamc@63 165 Section stream_eq.
adamc@63 166 Variable A : Set.
adamc@63 167
adamc@63 168 CoInductive stream_eq : stream A -> stream A -> Prop :=
adamc@63 169 | Stream_eq : forall h t1 t2,
adamc@63 170 stream_eq t1 t2
adamc@63 171 -> stream_eq (Cons h t1) (Cons h t2).
adamc@63 172 End stream_eq.
adamc@63 173
adamc@63 174 (** We say that two streams are equal if and only if they have the same heads and their tails are equal. We use the normal finite-syntactic equality for the heads, and we refer to our new equality recursively for the tails.
adamc@63 175
adamc@63 176 We can try restating the theorem with [stream_eq]. *)
adamc@63 177
adamc@63 178 Theorem ones_eq : stream_eq ones ones'.
adamc@63 179 (** Coq does not support tactical co-inductive proofs as well as it supports tactical inductive proofs. The usual starting point is the [cofix] tactic, which asks to structure this proof as a co-fixpoint. *)
adamc@63 180 cofix.
adamc@63 181 (** [[
adamc@63 182
adamc@63 183 ones_eq : stream_eq ones ones'
adamc@63 184 ============================
adamc@63 185 stream_eq ones ones'
adamc@63 186 ]] *)
adamc@63 187
adamc@63 188 (** It looks like this proof might be easier than we expected! *)
adamc@63 189
adamc@63 190 assumption.
adamc@63 191 (** [[
adamc@63 192
adamc@63 193 Proof completed. *)
adamc@63 194
adamc@63 195 (** Unfortunately, we are due for some disappointment in our victory lap. *)
adamc@63 196
adamc@63 197 (** [[
adamc@63 198 Qed.
adamc@63 199
adamc@63 200 Error:
adamc@63 201 Recursive definition of ones_eq is ill-formed.
adamc@63 202
adamc@63 203 In environment
adamc@63 204 ones_eq : stream_eq ones ones'
adamc@63 205
adamc@63 206 unguarded recursive call in "ones_eq" *)
adamc@63 207
adamc@63 208 (** Via the Curry-Howard correspondence, the same guardedness condition applies to our co-inductive proofs as to our co-inductive data structures. We should be grateful that this proof is rejected, because, if it were not, the same proof structure could be used to prove any co-inductive theorem vacuously, by direct appeal to itself!
adamc@63 209
adamc@63 210 Thinking about how Coq would generate a proof term from the proof script above, we see that the problem is that we are violating the first part of the guardedness condition. During our proofs, Coq can help us check whether we have yet gone wrong in this way. We can run the command [Guarded] in any context to see if it is possible to finish the proof in a way that will yield a properly guarded proof term.
adamc@63 211
adamc@63 212 [[
adamc@63 213 Guarded.
adamc@63 214
adamc@63 215 Running [Guarded] here gives us the same error message that we got when we tried to run [Qed]. In larger proofs, [Guarded] can be helpful in detecting problems %\textit{%#<i>#before#</i>#%}% we think we are ready to run [Qed].
adamc@63 216
adamc@63 217 We need to start the co-induction by applying one of [stream_eq]'s constructors. To do that, we need to know that both arguments to the predicate are [Cons]es. Informally, this is trivial, but [simpl] is not able to help us. *)
adamc@63 218
adamc@63 219 Undo.
adamc@63 220 simpl.
adamc@63 221 (** [[
adamc@63 222
adamc@63 223 ones_eq : stream_eq ones ones'
adamc@63 224 ============================
adamc@63 225 stream_eq ones ones'
adamc@63 226 ]] *)
adamc@63 227
adamc@63 228 (** It turns out that we are best served by proving an auxiliary lemma. *)
adamc@63 229 Abort.
adamc@63 230
adamc@63 231 (** First, we need to define a function that seems pointless on first glance. *)
adamc@63 232
adamc@63 233 Definition frob A (s : stream A) : stream A :=
adamc@63 234 match s with
adamc@63 235 | Cons h t => Cons h t
adamc@63 236 end.
adamc@63 237
adamc@63 238 (** Next, we need to prove a theorem that seems equally pointless. *)
adamc@63 239
adamc@63 240 Theorem frob_eq : forall A (s : stream A), s = frob s.
adamc@63 241 destruct s; reflexivity.
adamc@63 242 Qed.
adamc@63 243
adamc@63 244 (** But, miraculously, this theorem turns out to be just what we needed. *)
adamc@63 245
adamc@63 246 Theorem ones_eq : stream_eq ones ones'.
adamc@63 247 cofix.
adamc@63 248
adamc@63 249 (** We can use the theorem to rewrite the two streams. *)
adamc@63 250 rewrite (frob_eq ones).
adamc@63 251 rewrite (frob_eq ones').
adamc@63 252 (** [[
adamc@63 253
adamc@63 254 ones_eq : stream_eq ones ones'
adamc@63 255 ============================
adamc@63 256 stream_eq (frob ones) (frob ones')
adamc@63 257 ]] *)
adamc@63 258
adamc@63 259 (** Now [simpl] is able to reduce the streams. *)
adamc@63 260
adamc@63 261 simpl.
adamc@63 262 (** [[
adamc@63 263
adamc@63 264 ones_eq : stream_eq ones ones'
adamc@63 265 ============================
adamc@63 266 stream_eq (Cons 1 ones)
adamc@63 267 (Cons 1
adamc@63 268 ((cofix map (s : stream nat) : stream nat :=
adamc@63 269 match s with
adamc@63 270 | Cons h t => Cons (S h) (map t)
adamc@63 271 end) zeroes))
adamc@63 272 ]] *)
adamc@63 273
adamc@63 274 (** Since we have exposed the [Cons] structure of each stream, we can apply the constructor of [stream_eq]. *)
adamc@63 275
adamc@63 276 constructor.
adamc@63 277 (** [[
adamc@63 278
adamc@63 279 ones_eq : stream_eq ones ones'
adamc@63 280 ============================
adamc@63 281 stream_eq ones
adamc@63 282 ((cofix map (s : stream nat) : stream nat :=
adamc@63 283 match s with
adamc@63 284 | Cons h t => Cons (S h) (map t)
adamc@63 285 end) zeroes)
adamc@63 286 ]] *)
adamc@63 287
adamc@63 288 (** Now, modulo unfolding of the definition of [map], we have matched our assumption. *)
adamc@63 289 assumption.
adamc@63 290 Qed.
adamc@63 291
adamc@63 292 (** Why did this silly-looking trick help? The answer has to do with the constraints placed on Coq's evaluation rules by the need for termination. The [cofix]-related restriction that foiled our first attempt at using [simpl] is dual to a restriction for [fix]. In particular, an application of an anonymous [fix] only reduces when the top-level structure of the recursive argument is known. Otherwise, we would be unfolding the recursive definition ad infinitum.
adamc@63 293
adamc@63 294 Fixpoints only reduce when enough is known about the %\textit{%#<i>#definitions#</i>#%}% of their arguments. Dually, co-fixpoints only reduce when enough is known about %\textit{%#<i>#how their results will be used#</i>#%}%. In particular, a [cofix] is only expanded when it is the discriminee of a [match]. Rewriting with our superficially silly lemma wrapped new [match]es around the two [cofix]es, triggering reduction.
adamc@63 295
adamc@63 296 If [cofix]es reduced haphazardly, it would be easy to run into infinite loops in evaluation, since we are, after all, building infinite objects.
adamc@63 297
adamc@63 298 One common source of difficulty with co-inductive proofs is bad interaction with standard Coq automation machinery. If we try to prove [ones_eq'] with automation, like we have in previous inductive proofs, we get an invalid proof. *)
adamc@63 299
adamc@63 300 Theorem ones_eq' : stream_eq ones ones'.
adamc@63 301 cofix; crush.
adamc@63 302 (** [[
adamc@63 303
adamc@63 304 Guarded. *)
adamc@63 305 Abort.
adamc@63 306
adamc@63 307 (** The standard [auto] machinery sees that our goal matches an assumption and so applies that assumption, even though this violates guardedness. One usually starts a proof like this by [destruct]ing some parameter and running a custom tactic to figure out the first proof rule to apply for each case. Alternatively, there are tricks that can be played with "hiding" the co-inductive hypothesis. We will see examples of effective co-inductive proving in later chapters. *)