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author | Adam Chlipala <adam@chlipala.net> |
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date | Sun, 11 Nov 2012 18:17:23 -0500 |

parents | 1fd4109f7b31 |

children | ed829eaa91b2 |

rev | line source |
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adam@398 | 1 (* Copyright (c) 2008-2012, 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 |

adam@314 | 13 Require Import CpdtTactics. |

adamc@62 | 14 |

adam@422 | 15 Definition bad : unit := tt. |

adam@402 | 16 |

adamc@62 | 17 Set Implicit Arguments. |

adamc@62 | 18 (* end hide *) |

adamc@62 | 19 |

adamc@62 | 20 |

adamc@62 | 21 (** %\chapter{Infinite Data and Proofs}% *) |

adamc@62 | 22 |

adam@395 | 23 (** In lazy functional programming languages like %\index{Haskell}%Haskell, infinite data structures are everywhere%~\cite{whyfp}%. 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 | 24 |

adam@346 | 25 %\index{laziness}%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 | 26 |

adam@398 | 27 We spent some time in the last chapter discussing the %\index{Curry-Howard correspondence}%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 _inconsistent_. For an arbitrary proposition [P], we could write: |

adamc@202 | 28 [[ |

adamc@202 | 29 Fixpoint bad (u : unit) : P := bad u. |

adamc@205 | 30 ]] |

adamc@205 | 31 |

adam@442 | 32 This would leave us with [bad tt] as a proof of [P]. |

adamc@62 | 33 |

adamc@62 | 34 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 | 35 |

adam@422 | 36 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; several such approaches are surveyed in Chapter 7. This is a heavyweight solution, and so we would like to avoid it whenever possible. |

adamc@62 | 37 |

adam@402 | 38 Luckily, Coq has special support for a class of lazy data structures that happens to contain most examples found in Haskell. That mechanism,%\index{co-inductive types}% _co-inductive types_, is the subject of this chapter. *) |

adamc@62 | 39 |

adamc@62 | 40 |

adamc@62 | 41 (** * Computing with Infinite Data *) |

adamc@62 | 42 |

adam@398 | 43 (** Let us begin with the most basic type of infinite data, _streams_, or lazy lists.%\index{Vernacular commands!CoInductive}% *) |

adamc@62 | 44 |

adamc@62 | 45 Section stream. |

adam@351 | 46 Variable A : Type. |

adamc@62 | 47 |

adam@351 | 48 CoInductive stream : Type := |

adamc@62 | 49 | Cons : A -> stream -> stream. |

adamc@62 | 50 End stream. |

adamc@62 | 51 |

adam@422 | 52 (* begin hide *) |

adam@437 | 53 (* begin thide *) |

adam@422 | 54 CoInductive evilStream := Nil. |

adam@437 | 55 (* end thide *) |

adam@422 | 56 (* end hide *) |

adam@422 | 57 |

adamc@62 | 58 (** 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 | 59 |

adam@471 | 60 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 _use_ values of recursive inductive types effectively, here we find that we need%\index{co-recursive definitions}% _co-recursive definitions_ to _build_ values of co-inductive types effectively. |

adamc@62 | 61 |

adam@346 | 62 We can define a stream consisting only of zeroes.%\index{Vernacular commands!CoFixpoint}% *) |

adamc@62 | 63 |

adamc@62 | 64 CoFixpoint zeroes : stream nat := Cons 0 zeroes. |

adamc@62 | 65 |

adam@346 | 66 (* EX: Define a stream that alternates between [true] and [false]. *) |

adam@346 | 67 (* begin thide *) |

adam@346 | 68 |

adamc@62 | 69 (** We can also define a stream that alternates between [true] and [false]. *) |

adamc@62 | 70 |

adam@346 | 71 CoFixpoint trues_falses : stream bool := Cons true falses_trues |

adam@346 | 72 with falses_trues : stream bool := Cons false trues_falses. |

adam@346 | 73 (* end thide *) |

adamc@62 | 74 |

adamc@62 | 75 (** 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 | 76 |

adam@348 | 77 (* EX: Define a function to calculate a finite approximation of a stream, to a particular length. *) |

adam@346 | 78 (* begin thide *) |

adam@346 | 79 |

adamc@211 | 80 Fixpoint approx A (s : stream A) (n : nat) : list A := |

adamc@62 | 81 match n with |

adamc@62 | 82 | O => nil |

adamc@62 | 83 | S n' => |

adamc@62 | 84 match s with |

adamc@62 | 85 | Cons h t => h :: approx t n' |

adamc@62 | 86 end |

adamc@62 | 87 end. |

adamc@62 | 88 |

adamc@62 | 89 Eval simpl in approx zeroes 10. |

adamc@211 | 90 (** %\vspace{-.15in}% [[ |

adamc@62 | 91 = 0 :: 0 :: 0 :: 0 :: 0 :: 0 :: 0 :: 0 :: 0 :: 0 :: nil |

adamc@62 | 92 : list nat |

adam@302 | 93 ]] |

adam@302 | 94 *) |

adamc@211 | 95 |

adam@346 | 96 Eval simpl in approx trues_falses 10. |

adamc@211 | 97 (** %\vspace{-.15in}% [[ |

adamc@62 | 98 = true |

adamc@62 | 99 :: false |

adamc@62 | 100 :: true |

adamc@62 | 101 :: false |

adamc@62 | 102 :: true :: false :: true :: false :: true :: false :: nil |

adamc@62 | 103 : list bool |

adam@346 | 104 ]] |

adam@346 | 105 *) |

adamc@62 | 106 |

adam@349 | 107 (* end thide *) |

adamc@62 | 108 |

adam@402 | 109 (* begin hide *) |

adam@437 | 110 (* begin thide *) |

adam@402 | 111 Definition looper := 0. |

adam@437 | 112 (* end thide *) |

adam@402 | 113 (* end hide *) |

adam@402 | 114 |

adam@398 | 115 (** 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 _consume_ values of inductive types, with restrictions on which _arguments_ may be passed in recursive calls. Dually, co-fixpoints _produce_ values of co-inductive types, with restrictions on what may be done with the _results_ of co-recursive calls. |

adamc@62 | 116 |

adam@402 | 117 The restriction for co-inductive types shows up as the%\index{guardedness condition}% _guardedness condition_. First, consider this stream definition, which would be legal in Haskell. |

adamc@62 | 118 [[ |

adamc@62 | 119 CoFixpoint looper : stream nat := looper. |

adam@346 | 120 ]] |

adamc@205 | 121 |

adam@346 | 122 << |

adamc@62 | 123 Error: |

adamc@62 | 124 Recursive definition of looper is ill-formed. |

adamc@62 | 125 In environment |

adamc@62 | 126 looper : stream nat |

adamc@62 | 127 |

adamc@62 | 128 unguarded recursive call in "looper" |

adam@346 | 129 >> |

adamc@205 | 130 |

adam@398 | 131 The rule we have run afoul of here is that _every co-recursive call must be guarded by a constructor_; 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 | 132 |

adam@346 | 133 Some familiar functions are easy to write in co-recursive fashion. *) |

adamc@62 | 134 |

adamc@62 | 135 Section map. |

adam@351 | 136 Variables A B : Type. |

adamc@62 | 137 Variable f : A -> B. |

adamc@62 | 138 |

adamc@62 | 139 CoFixpoint map (s : stream A) : stream B := |

adamc@62 | 140 match s with |

adamc@62 | 141 | Cons h t => Cons (f h) (map t) |

adamc@62 | 142 end. |

adamc@62 | 143 End map. |

adamc@62 | 144 |

adam@402 | 145 (* begin hide *) |

adam@437 | 146 (* begin thide *) |

adam@402 | 147 Definition filter := 0. |

adam@437 | 148 (* end thide *) |

adam@402 | 149 (* end hide *) |

adam@402 | 150 |

adam@402 | 151 (** 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 the guardedness condition. |

adamc@62 | 152 |

adam@398 | 153 The implications of the condition can be subtle. To illustrate how, we start off with another co-recursive function definition that _is_ legal. The function [interleave] takes two streams and produces a new stream that alternates between their elements. *) |

adamc@62 | 154 |

adamc@62 | 155 Section interleave. |

adam@351 | 156 Variable A : Type. |

adamc@62 | 157 |

adamc@62 | 158 CoFixpoint interleave (s1 s2 : stream A) : stream A := |

adamc@62 | 159 match s1, s2 with |

adamc@62 | 160 | Cons h1 t1, Cons h2 t2 => Cons h1 (Cons h2 (interleave t1 t2)) |

adamc@62 | 161 end. |

adamc@62 | 162 End interleave. |

adamc@62 | 163 |

adamc@62 | 164 (** Now say we want to write a weird stuttering version of [map] that repeats elements in a particular way, based on interleaving. *) |

adamc@62 | 165 |

adamc@62 | 166 Section map'. |

adam@351 | 167 Variables A B : Type. |

adamc@62 | 168 Variable f : A -> B. |

adamc@68 | 169 (* begin thide *) |

adam@440 | 170 (** %\vspace{-.15in}%[[ |

adamc@62 | 171 CoFixpoint map' (s : stream A) : stream B := |

adamc@62 | 172 match s with |

adam@346 | 173 | Cons h t => interleave (Cons (f h) (map' t)) (Cons (f h) (map' t)) |

adamc@68 | 174 end. |

adamc@205 | 175 ]] |

adam@440 | 176 %\vspace{-.15in}%We get another error message about an unguarded recursive call. *) |

adamc@62 | 177 |

adam@346 | 178 End map'. |

adam@346 | 179 |

adam@475 | 180 (** What is going wrong here? Imagine that, instead of [interleave], we had called some other, less well-behaved function on streams. Here is one simpler example demonstrating the essential pitfall. We start by defining a standard function for taking the tail of a stream. Since streams are infinite, this operation is total. *) |

adam@346 | 181 |

adam@346 | 182 Definition tl A (s : stream A) : stream A := |

adam@346 | 183 match s with |

adam@346 | 184 | Cons _ s' => s' |

adam@346 | 185 end. |

adam@346 | 186 |

adam@346 | 187 (** Coq rejects the following definition that uses [tl]. |

adam@346 | 188 [[ |

adam@346 | 189 CoFixpoint bad : stream nat := tl (Cons 0 bad). |

adam@346 | 190 ]] |

adam@346 | 191 |

adam@422 | 192 Imagine that Coq had accepted our definition, and consider how we might evaluate [approx bad 1]. We would be trying to calculate the first element in the stream [bad]. However, it is not hard to see that the definition of [bad] "begs the question": unfolding the definition of [tl], we see that we essentially say "define [bad] to equal itself"! Of course such an equation admits no single well-defined solution, which does not fit well with the determinism of Gallina reduction. |

adam@346 | 193 |

adam@471 | 194 Coq's complete rule for co-recursive definitions includes not just the basic guardedness condition, but also a requirement about where co-recursive calls may occur. In particular, a co-recursive call must be a direct argument to a constructor, _nested only inside of other constructor calls or [fun] or [match] expressions_. In the definition of [bad], we erroneously nested the co-recursive call inside a call to [tl], and we nested inside a call to [interleave] in the definition of [map']. |

adam@471 | 195 |

adam@471 | 196 Coq helps the user out a little by performing the guardedness check after using computation to simplify terms. For instance, any co-recursive function definition can be expanded by inserting extra calls to an identity function, and this change preserves guardedness. However, in other cases computational simplification can reveal why definitions are dangerous. Consider what happens when we inline the definition of [tl] in [bad]: |

adam@346 | 197 [[ |

adam@346 | 198 CoFixpoint bad : stream nat := bad. |

adam@346 | 199 ]] |

adam@471 | 200 This is the same looping definition we rejected earlier. A similar inlining process reveals an alternate view on our failed definition of [map']: |

adam@346 | 201 [[ |

adam@346 | 202 CoFixpoint map' (s : stream A) : stream B := |

adam@346 | 203 match s with |

adam@346 | 204 | Cons h t => Cons (f h) (Cons (f h) (interleave (map' t) (map' t))) |

adam@346 | 205 end. |

adam@346 | 206 ]] |

adam@346 | 207 Clearly in this case the [map'] calls are not immediate arguments to constructors, so we violate the guardedness condition. *) |

adamc@68 | 208 (* end thide *) |

adamc@211 | 209 |

adam@402 | 210 (** A more interesting question is why that condition is the right one. We can make an intuitive argument that the original [map'] definition is perfectly reasonable and denotes a well-understood transformation on streams, such that every output would behave properly with [approx]. The guardedness condition is an example of a syntactic check for%\index{productivity}% _productivity_ of co-recursive definitions. A productive definition can be thought of as one whose outputs can be forced in finite time to any finite approximation level, as with [approx]. If we replaced the guardedness condition with more involved checks, we might be able to detect and allow a broader range of productive definitions. However, mistakes in these checks could cause inconsistency, and programmers would need to understand the new, more complex checks. Coq's design strikes a balance between consistency and simplicity with its choice of guard condition, though we can imagine other worthwhile balances being struck, too. *) |

adamc@62 | 211 |

adamc@63 | 212 |

adamc@63 | 213 (** * Infinite Proofs *) |

adamc@63 | 214 |

adamc@63 | 215 (** 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 | 216 |

adamc@63 | 217 CoFixpoint ones : stream nat := Cons 1 ones. |

adamc@63 | 218 Definition ones' := map S zeroes. |

adamc@63 | 219 |

adamc@63 | 220 (** The obvious statement of the equality is this: *) |

adamc@63 | 221 |

adamc@63 | 222 Theorem ones_eq : ones = ones'. |

adamc@63 | 223 |

adam@422 | 224 (* begin hide *) |

adam@437 | 225 (* begin thide *) |

adam@422 | 226 Definition foo := @eq. |

adam@437 | 227 (* end thide *) |

adam@422 | 228 (* end hide *) |

adam@422 | 229 |

adamc@63 | 230 (** 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@68 | 231 (* begin thide *) |

adamc@211 | 232 |

adamc@63 | 233 Abort. |

adamc@63 | 234 |

adam@398 | 235 (** Co-inductive datatypes make sense by analogy from Haskell. What we need now is a _co-inductive proposition_. 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 | 236 |

adam@346 | 237 We are ready for our first %\index{co-inductive predicates}%co-inductive predicate. *) |

adamc@63 | 238 |

adamc@63 | 239 Section stream_eq. |

adam@351 | 240 Variable A : Type. |

adamc@63 | 241 |

adamc@63 | 242 CoInductive stream_eq : stream A -> stream A -> Prop := |

adamc@63 | 243 | Stream_eq : forall h t1 t2, |

adamc@63 | 244 stream_eq t1 t2 |

adamc@63 | 245 -> stream_eq (Cons h t1) (Cons h t2). |

adamc@63 | 246 End stream_eq. |

adamc@63 | 247 |

adamc@63 | 248 (** 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 | 249 |

adamc@63 | 250 We can try restating the theorem with [stream_eq]. *) |

adamc@63 | 251 |

adamc@63 | 252 Theorem ones_eq : stream_eq ones ones'. |

adamc@63 | 253 (** 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@211 | 254 |

adamc@63 | 255 cofix. |

adamc@63 | 256 (** [[ |

adamc@63 | 257 ones_eq : stream_eq ones ones' |

adamc@63 | 258 ============================ |

adamc@63 | 259 stream_eq ones ones' |

adamc@211 | 260 |

adamc@211 | 261 ]] |

adamc@63 | 262 |

adamc@211 | 263 It looks like this proof might be easier than we expected! *) |

adamc@63 | 264 |

adamc@63 | 265 assumption. |

adam@422 | 266 (** << |

adamc@211 | 267 Proof completed. |

adam@422 | 268 >> |

adamc@63 | 269 |

adamc@211 | 270 Unfortunately, we are due for some disappointment in our victory lap. |

adamc@211 | 271 [[ |

adamc@63 | 272 Qed. |

adam@346 | 273 ]] |

adamc@63 | 274 |

adam@346 | 275 << |

adamc@63 | 276 Error: |

adamc@63 | 277 Recursive definition of ones_eq is ill-formed. |

adamc@63 | 278 |

adamc@63 | 279 In environment |

adamc@63 | 280 ones_eq : stream_eq ones ones' |

adamc@63 | 281 |

adamc@205 | 282 unguarded recursive call in "ones_eq" |

adam@346 | 283 >> |

adamc@205 | 284 |

adamc@211 | 285 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 | 286 |

adam@347 | 287 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 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.%\index{Vernacular commands!Guarded}% |

adamc@63 | 288 [[ |

adamc@63 | 289 Guarded. |

adamc@205 | 290 ]] |

adamc@205 | 291 |

adam@398 | 292 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 _before_ we think we are ready to run [Qed]. |

adamc@63 | 293 |

adam@281 | 294 We need to start the co-induction by applying [stream_eq]'s constructor. 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 | 295 |

adamc@63 | 296 Undo. |

adamc@63 | 297 simpl. |

adamc@63 | 298 (** [[ |

adamc@63 | 299 ones_eq : stream_eq ones ones' |

adamc@63 | 300 ============================ |

adamc@63 | 301 stream_eq ones ones' |

adamc@211 | 302 |

adamc@211 | 303 ]] |

adamc@63 | 304 |

adamc@211 | 305 It turns out that we are best served by proving an auxiliary lemma. *) |

adamc@211 | 306 |

adamc@63 | 307 Abort. |

adamc@63 | 308 |

adam@450 | 309 (** First, we need to define a function that seems pointless at first glance. *) |

adamc@63 | 310 |

adamc@63 | 311 Definition frob A (s : stream A) : stream A := |

adamc@63 | 312 match s with |

adamc@63 | 313 | Cons h t => Cons h t |

adamc@63 | 314 end. |

adamc@63 | 315 |

adamc@63 | 316 (** Next, we need to prove a theorem that seems equally pointless. *) |

adamc@63 | 317 |

adamc@63 | 318 Theorem frob_eq : forall A (s : stream A), s = frob s. |

adamc@63 | 319 destruct s; reflexivity. |

adamc@63 | 320 Qed. |

adamc@63 | 321 |

adamc@63 | 322 (** But, miraculously, this theorem turns out to be just what we needed. *) |

adamc@63 | 323 |

adamc@63 | 324 Theorem ones_eq : stream_eq ones ones'. |

adamc@63 | 325 cofix. |

adamc@63 | 326 |

adamc@63 | 327 (** We can use the theorem to rewrite the two streams. *) |

adamc@211 | 328 |

adamc@63 | 329 rewrite (frob_eq ones). |

adamc@63 | 330 rewrite (frob_eq ones'). |

adamc@63 | 331 (** [[ |

adamc@63 | 332 ones_eq : stream_eq ones ones' |

adamc@63 | 333 ============================ |

adamc@63 | 334 stream_eq (frob ones) (frob ones') |

adamc@211 | 335 |

adamc@211 | 336 ]] |

adamc@63 | 337 |

adamc@211 | 338 Now [simpl] is able to reduce the streams. *) |

adamc@63 | 339 |

adamc@63 | 340 simpl. |

adamc@63 | 341 (** [[ |

adamc@63 | 342 ones_eq : stream_eq ones ones' |

adamc@63 | 343 ============================ |

adamc@63 | 344 stream_eq (Cons 1 ones) |

adamc@63 | 345 (Cons 1 |

adamc@63 | 346 ((cofix map (s : stream nat) : stream nat := |

adamc@63 | 347 match s with |

adamc@63 | 348 | Cons h t => Cons (S h) (map t) |

adamc@63 | 349 end) zeroes)) |

adamc@211 | 350 |

adamc@211 | 351 ]] |

adamc@63 | 352 |

adam@422 | 353 Note the [cofix] notation for anonymous co-recursion, which is analogous to the [fix] notation we have already seen for recursion. Since we have exposed the [Cons] structure of each stream, we can apply the constructor of [stream_eq]. *) |

adamc@63 | 354 |

adamc@63 | 355 constructor. |

adamc@63 | 356 (** [[ |

adamc@63 | 357 ones_eq : stream_eq ones ones' |

adamc@63 | 358 ============================ |

adamc@63 | 359 stream_eq ones |

adamc@63 | 360 ((cofix map (s : stream nat) : stream nat := |

adamc@63 | 361 match s with |

adamc@63 | 362 | Cons h t => Cons (S h) (map t) |

adamc@63 | 363 end) zeroes) |

adamc@211 | 364 |

adamc@211 | 365 ]] |

adamc@63 | 366 |

adamc@211 | 367 Now, modulo unfolding of the definition of [map], we have matched our assumption. *) |

adamc@211 | 368 |

adamc@63 | 369 assumption. |

adamc@63 | 370 Qed. |

adamc@63 | 371 |

adamc@63 | 372 (** 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 | 373 |

adam@398 | 374 Fixpoints only reduce when enough is known about the _definitions_ of their arguments. Dually, co-fixpoints only reduce when enough is known about _how their results will be used_. 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 | 375 |

adamc@63 | 376 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 | 377 |

adamc@63 | 378 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 | 379 |

adamc@63 | 380 Theorem ones_eq' : stream_eq ones ones'. |

adamc@63 | 381 cofix; crush. |

adam@440 | 382 (** %\vspace{-.25in}%[[ |

adamc@205 | 383 Guarded. |

adam@302 | 384 ]] |

adam@440 | 385 %\vspace{-.25in}% |

adam@302 | 386 *) |

adamc@63 | 387 Abort. |

adam@346 | 388 |

adam@471 | 389 (** The standard [auto] machinery sees that our goal matches an assumption and so applies that assumption, even though this violates guardedness. A correct proof strategy for a theorem like this usually starts 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. |

adam@346 | 390 |

adam@346 | 391 %\medskip% |

adam@346 | 392 |

adam@402 | 393 Must we always be cautious with automation in proofs by co-induction? Induction seems to have dual versions of the same pitfalls inherent in it, and yet we avoid those pitfalls by encapsulating safe Curry-Howard recursion schemes inside named induction principles. It turns out that we can usually do the same with%\index{co-induction principles}% _co-induction principles_. Let us take that tack here, so that we can arrive at an [induction x; crush]-style proof for [ones_eq']. |

adam@346 | 394 |

adam@398 | 395 An induction principle is parameterized over a predicate characterizing what we mean to prove, _as a function of the inductive fact that we already know_. Dually, a co-induction principle ought to be parameterized over a predicate characterizing what we mean to prove, _as a function of the arguments to the co-inductive predicate that we are trying to prove_. |

adam@346 | 396 |

adam@346 | 397 To state a useful principle for [stream_eq], it will be useful first to define the stream head function. *) |

adam@346 | 398 |

adam@346 | 399 Definition hd A (s : stream A) : A := |

adam@346 | 400 match s with |

adam@346 | 401 | Cons x _ => x |

adam@346 | 402 end. |

adam@346 | 403 |

adam@346 | 404 (** Now we enter a section for the co-induction principle, based on %\index{Park's principle}%Park's principle as introduced in a tutorial by Gim%\'%enez%~\cite{IT}%. *) |

adam@346 | 405 |

adam@346 | 406 Section stream_eq_coind. |

adam@351 | 407 Variable A : Type. |

adam@346 | 408 Variable R : stream A -> stream A -> Prop. |

adam@475 | 409 |

adam@471 | 410 (** This relation generalizes the theorem we want to prove, defining a set of pairs of streams that we must eventually prove contains the particular pair we care about. *) |

adam@346 | 411 |

adam@346 | 412 Hypothesis Cons_case_hd : forall s1 s2, R s1 s2 -> hd s1 = hd s2. |

adam@346 | 413 Hypothesis Cons_case_tl : forall s1 s2, R s1 s2 -> R (tl s1) (tl s2). |

adam@475 | 414 |

adam@422 | 415 (** Two hypotheses characterize what makes a good choice of [R]: it enforces equality of stream heads, and it is %``%#<i>#hereditary#</i>#%''% in the sense that an [R] stream pair passes on "[R]-ness" to its tails. An established technical term for such a relation is%\index{bisimulation}% _bisimulation_. *) |

adam@346 | 416 |

adam@346 | 417 (** Now it is straightforward to prove the principle, which says that any stream pair in [R] is equal. The reader may wish to step through the proof script to see what is going on. *) |

adam@392 | 418 |

adam@346 | 419 Theorem stream_eq_coind : forall s1 s2, R s1 s2 -> stream_eq s1 s2. |

adam@346 | 420 cofix; destruct s1; destruct s2; intro. |

adam@346 | 421 generalize (Cons_case_hd H); intro Heq; simpl in Heq; rewrite Heq. |

adam@346 | 422 constructor. |

adam@346 | 423 apply stream_eq_coind. |

adam@346 | 424 apply (Cons_case_tl H). |

adam@346 | 425 Qed. |

adam@346 | 426 End stream_eq_coind. |

adam@346 | 427 |

adam@346 | 428 (** To see why this proof is guarded, we can print it and verify that the one co-recursive call is an immediate argument to a constructor. *) |

adam@392 | 429 |

adam@346 | 430 Print stream_eq_coind. |

adam@346 | 431 |

adam@346 | 432 (** We omit the output and proceed to proving [ones_eq''] again. The only bit of ingenuity is in choosing [R], and in this case the most restrictive predicate works. *) |

adam@346 | 433 |

adam@346 | 434 Theorem ones_eq'' : stream_eq ones ones'. |

adam@346 | 435 apply (stream_eq_coind (fun s1 s2 => s1 = ones /\ s2 = ones')); crush. |

adam@346 | 436 Qed. |

adam@346 | 437 |

adam@346 | 438 (** Note that this proof achieves the proper reduction behavior via [hd] and [tl], rather than [frob] as in the last proof. All three functions pattern match on their arguments, catalyzing computation steps. |

adam@346 | 439 |

adam@346 | 440 Compared to the inductive proofs that we are used to, it still seems unsatisfactory that we had to write out a choice of [R] in the last proof. An alternate is to capture a common pattern of co-recursion in a more specialized co-induction principle. For the current example, that pattern is: prove [stream_eq s1 s2] where [s1] and [s2] are defined as their own tails. *) |

adam@346 | 441 |

adam@346 | 442 Section stream_eq_loop. |

adam@351 | 443 Variable A : Type. |

adam@346 | 444 Variables s1 s2 : stream A. |

adam@346 | 445 |

adam@346 | 446 Hypothesis Cons_case_hd : hd s1 = hd s2. |

adam@346 | 447 Hypothesis loop1 : tl s1 = s1. |

adam@346 | 448 Hypothesis loop2 : tl s2 = s2. |

adam@346 | 449 |

adam@346 | 450 (** The proof of the principle includes a choice of [R], so that we no longer need to make such choices thereafter. *) |

adam@346 | 451 |

adam@346 | 452 Theorem stream_eq_loop : stream_eq s1 s2. |

adam@346 | 453 apply (stream_eq_coind (fun s1' s2' => s1' = s1 /\ s2' = s2)); crush. |

adam@346 | 454 Qed. |

adam@346 | 455 End stream_eq_loop. |

adam@346 | 456 |

adam@346 | 457 Theorem ones_eq''' : stream_eq ones ones'. |

adam@346 | 458 apply stream_eq_loop; crush. |

adam@346 | 459 Qed. |

adamc@68 | 460 (* end thide *) |

adamc@63 | 461 |

adam@435 | 462 (** Let us put [stream_eq_coind] through its paces a bit more, considering two different ways to compute infinite streams of all factorial values. First, we import the [fact] factorial function from the standard library. *) |

adam@346 | 463 |

adam@346 | 464 Require Import Arith. |

adam@346 | 465 Print fact. |

adam@346 | 466 (** %\vspace{-.15in}%[[ |

adam@346 | 467 fact = |

adam@346 | 468 fix fact (n : nat) : nat := |

adam@346 | 469 match n with |

adam@346 | 470 | 0 => 1 |

adam@346 | 471 | S n0 => S n0 * fact n0 |

adam@346 | 472 end |

adam@346 | 473 : nat -> nat |

adam@346 | 474 ]] |

adam@346 | 475 *) |

adam@346 | 476 |

adam@346 | 477 (** The simplest way to compute the factorial stream involves calling [fact] afresh at each position. *) |

adam@346 | 478 |

adam@346 | 479 CoFixpoint fact_slow' (n : nat) := Cons (fact n) (fact_slow' (S n)). |

adam@346 | 480 Definition fact_slow := fact_slow' 1. |

adam@346 | 481 |

adam@346 | 482 (** A more clever, optimized method maintains an accumulator of the previous factorial, so that each new entry can be computed with a single multiplication. *) |

adam@346 | 483 |

adam@346 | 484 CoFixpoint fact_iter' (cur acc : nat) := Cons acc (fact_iter' (S cur) (acc * cur)). |

adam@346 | 485 Definition fact_iter := fact_iter' 2 1. |

adam@346 | 486 |

adam@346 | 487 (** We can verify that the streams are equal up to particular finite bounds. *) |

adam@346 | 488 |

adam@346 | 489 Eval simpl in approx fact_iter 5. |

adam@346 | 490 (** %\vspace{-.15in}%[[ |

adam@346 | 491 = 1 :: 2 :: 6 :: 24 :: 120 :: nil |

adam@346 | 492 : list nat |

adam@346 | 493 ]] |

adam@346 | 494 *) |

adam@346 | 495 Eval simpl in approx fact_slow 5. |

adam@346 | 496 (** %\vspace{-.15in}%[[ |

adam@346 | 497 = 1 :: 2 :: 6 :: 24 :: 120 :: nil |

adam@346 | 498 : list nat |

adam@346 | 499 ]] |

adam@346 | 500 |

adam@471 | 501 Now, to prove that the two versions are equivalent, it is helpful to prove (and add as a proof hint) a quick lemma about the computational behavior of [fact]. (I intentionally skip explaining its proof at this point.) *) |

adam@346 | 502 |

adam@346 | 503 (* begin thide *) |

adam@346 | 504 Lemma fact_def : forall x n, |

adam@346 | 505 fact_iter' x (fact n * S n) = fact_iter' x (fact (S n)). |

adam@346 | 506 simpl; intros; f_equal; ring. |

adam@346 | 507 Qed. |

adam@346 | 508 |

adam@346 | 509 Hint Resolve fact_def. |

adam@346 | 510 |

adam@346 | 511 (** With the hint added, it is easy to prove an auxiliary lemma relating [fact_iter'] and [fact_slow']. The key bit of ingenuity is introduction of an existential quantifier for the shared parameter [n]. *) |

adam@346 | 512 |

adam@346 | 513 Lemma fact_eq' : forall n, stream_eq (fact_iter' (S n) (fact n)) (fact_slow' n). |

adam@346 | 514 intro; apply (stream_eq_coind (fun s1 s2 => exists n, s1 = fact_iter' (S n) (fact n) |

adam@346 | 515 /\ s2 = fact_slow' n)); crush; eauto. |

adam@346 | 516 Qed. |

adam@346 | 517 |

adam@346 | 518 (** The final theorem is a direct corollary of [fact_eq']. *) |

adam@346 | 519 |

adam@346 | 520 Theorem fact_eq : stream_eq fact_iter fact_slow. |

adam@346 | 521 apply fact_eq'. |

adam@346 | 522 Qed. |

adam@346 | 523 |

adam@346 | 524 (** As in the case of [ones_eq'], we may be unsatisfied that we needed to write down a choice of [R] that seems to duplicate information already present in a lemma statement. We can facilitate a simpler proof by defining a co-induction principle specialized to goals that begin with single universal quantifiers, and the strategy can be extended in a straightforward way to principles for other counts of quantifiers. (Our [stream_eq_loop] principle is effectively the instantiation of this technique to zero quantifiers.) *) |

adam@346 | 525 |

adam@346 | 526 Section stream_eq_onequant. |

adam@351 | 527 Variables A B : Type. |

adam@346 | 528 (** We have the types [A], the domain of the one quantifier; and [B], the type of data found in the streams. *) |

adam@346 | 529 |

adam@346 | 530 Variables f g : A -> stream B. |

adam@346 | 531 (** The two streams we compare must be of the forms [f x] and [g x], for some shared [x]. Note that this falls out naturally when [x] is a shared universally quantified variable in a lemma statement. *) |

adam@346 | 532 |

adam@346 | 533 Hypothesis Cons_case_hd : forall x, hd (f x) = hd (g x). |

adam@346 | 534 Hypothesis Cons_case_tl : forall x, exists y, tl (f x) = f y /\ tl (g x) = g y. |

adam@346 | 535 (** These conditions are inspired by the bisimulation requirements, with a more general version of the [R] choice we made for [fact_eq'] inlined into the hypotheses of [stream_eq_coind]. *) |

adam@346 | 536 |

adam@346 | 537 Theorem stream_eq_onequant : forall x, stream_eq (f x) (g x). |

adam@346 | 538 intro; apply (stream_eq_coind (fun s1 s2 => exists x, s1 = f x /\ s2 = g x)); crush; eauto. |

adam@346 | 539 Qed. |

adam@346 | 540 End stream_eq_onequant. |

adam@346 | 541 |

adam@346 | 542 Lemma fact_eq'' : forall n, stream_eq (fact_iter' (S n) (fact n)) (fact_slow' n). |

adam@346 | 543 apply stream_eq_onequant; crush; eauto. |

adam@346 | 544 Qed. |

adam@346 | 545 |

adam@346 | 546 (** We have arrived at one of our customary automated proofs, thanks to the new principle. *) |

adam@346 | 547 (* end thide *) |

adamc@64 | 548 |

adamc@64 | 549 |

adamc@64 | 550 (** * Simple Modeling of Non-Terminating Programs *) |

adamc@64 | 551 |

adam@402 | 552 (** We close the chapter with a quick motivating example for more complex uses of co-inductive types. We will define a co-inductive semantics for a simple imperative programming language and use that semantics to prove the correctness of a trivial optimization that removes spurious additions by 0. We follow the technique of%\index{co-inductive big-step operational semantics}% _co-inductive big-step operational semantics_ %\cite{BigStep}%. |

adamc@64 | 553 |

adam@471 | 554 We define a suggestive synonym for [nat], as we will consider programs over infinitely many variables, represented as [nat]s. *) |

adamc@211 | 555 |

adam@347 | 556 Definition var := nat. |

adamc@64 | 557 |

adam@436 | 558 (** We define a type [vars] of maps from variables to values. To define a function [set] for setting a variable's value in a map, we use the standard library function [beq_nat] for comparing natural numbers. *) |

adamc@64 | 559 |

adam@347 | 560 Definition vars := var -> nat. |

adam@347 | 561 Definition set (vs : vars) (v : var) (n : nat) : vars := |

adam@347 | 562 fun v' => if beq_nat v v' then n else vs v'. |

adamc@67 | 563 |

adam@347 | 564 (** We define a simple arithmetic expression language with variables, and we give it a semantics via an interpreter. *) |

adamc@67 | 565 |

adam@347 | 566 Inductive exp : Set := |

adam@347 | 567 | Const : nat -> exp |

adam@347 | 568 | Var : var -> exp |

adam@347 | 569 | Plus : exp -> exp -> exp. |

adamc@64 | 570 |

adam@347 | 571 Fixpoint evalExp (vs : vars) (e : exp) : nat := |

adam@347 | 572 match e with |

adam@347 | 573 | Const n => n |

adam@347 | 574 | Var v => vs v |

adam@347 | 575 | Plus e1 e2 => evalExp vs e1 + evalExp vs e2 |

adam@347 | 576 end. |

adamc@64 | 577 |

adam@422 | 578 (** Finally, we define a language of commands. It includes variable assignment, sequencing, and a <<while>> form that repeats as long as its test expression evaluates to a nonzero value. *) |

adamc@64 | 579 |

adam@347 | 580 Inductive cmd : Set := |

adam@347 | 581 | Assign : var -> exp -> cmd |

adam@347 | 582 | Seq : cmd -> cmd -> cmd |

adam@347 | 583 | While : exp -> cmd -> cmd. |

adamc@64 | 584 |

adam@471 | 585 (** We could define an inductive relation to characterize the results of command evaluation. However, such a relation would not capture _nonterminating_ executions. With a co-inductive relation, we can capture both cases. The parameters of the relation are an initial state, a command, and a final state. A program that does not terminate in a particular initial state is related to _any_ final state. For more realistic languages than this one, it is often possible for programs to _crash_, in which case a semantics would generally relate their executions to no final states; so relating safely non-terminating programs to all final states provides a crucial distinction. *) |

adamc@67 | 586 |

adam@347 | 587 CoInductive evalCmd : vars -> cmd -> vars -> Prop := |

adam@347 | 588 | EvalAssign : forall vs v e, evalCmd vs (Assign v e) (set vs v (evalExp vs e)) |

adam@347 | 589 | EvalSeq : forall vs1 vs2 vs3 c1 c2, evalCmd vs1 c1 vs2 |

adam@347 | 590 -> evalCmd vs2 c2 vs3 |

adam@347 | 591 -> evalCmd vs1 (Seq c1 c2) vs3 |

adam@347 | 592 | EvalWhileFalse : forall vs e c, evalExp vs e = 0 |

adam@347 | 593 -> evalCmd vs (While e c) vs |

adam@347 | 594 | EvalWhileTrue : forall vs1 vs2 vs3 e c, evalExp vs1 e <> 0 |

adam@347 | 595 -> evalCmd vs1 c vs2 |

adam@347 | 596 -> evalCmd vs2 (While e c) vs3 |

adam@347 | 597 -> evalCmd vs1 (While e c) vs3. |

adam@347 | 598 |

adam@347 | 599 (** Having learned our lesson in the last section, before proceeding, we build a co-induction principle for [evalCmd]. *) |

adam@347 | 600 |

adam@347 | 601 Section evalCmd_coind. |

adam@347 | 602 Variable R : vars -> cmd -> vars -> Prop. |

adam@347 | 603 |

adam@347 | 604 Hypothesis AssignCase : forall vs1 vs2 v e, R vs1 (Assign v e) vs2 |

adam@347 | 605 -> vs2 = set vs1 v (evalExp vs1 e). |

adam@347 | 606 |

adam@347 | 607 Hypothesis SeqCase : forall vs1 vs3 c1 c2, R vs1 (Seq c1 c2) vs3 |

adam@347 | 608 -> exists vs2, R vs1 c1 vs2 /\ R vs2 c2 vs3. |

adam@347 | 609 |

adam@347 | 610 Hypothesis WhileCase : forall vs1 vs3 e c, R vs1 (While e c) vs3 |

adam@347 | 611 -> (evalExp vs1 e = 0 /\ vs3 = vs1) |

adam@347 | 612 \/ exists vs2, evalExp vs1 e <> 0 /\ R vs1 c vs2 /\ R vs2 (While e c) vs3. |

adam@347 | 613 |

adam@402 | 614 (** The proof is routine. We make use of a form of %\index{tactics!destruct}%[destruct] that takes an%\index{intro pattern}% _intro pattern_ in an [as] clause. These patterns control how deeply we break apart the components of an inductive value, and we refer the reader to the Coq manual for more details. *) |

adam@347 | 615 |

adam@347 | 616 Theorem evalCmd_coind : forall vs1 c vs2, R vs1 c vs2 -> evalCmd vs1 c vs2. |

adam@347 | 617 cofix; intros; destruct c. |

adam@347 | 618 rewrite (AssignCase H); constructor. |

adam@347 | 619 destruct (SeqCase H) as [? [? ?]]; econstructor; eauto. |

adam@478 | 620 destruct (WhileCase H) as [[? ?] | [? [? [? ?]]]]; subst; econstructor; eauto. |

adam@347 | 621 Qed. |

adam@347 | 622 End evalCmd_coind. |

adam@347 | 623 |

adam@347 | 624 (** Now that we have a co-induction principle, we should use it to prove something! Our example is a trivial program optimizer that finds places to replace [0 + e] with [e]. *) |

adam@347 | 625 |

adam@347 | 626 Fixpoint optExp (e : exp) : exp := |

adam@347 | 627 match e with |

adam@347 | 628 | Plus (Const 0) e => optExp e |

adam@347 | 629 | Plus e1 e2 => Plus (optExp e1) (optExp e2) |

adam@347 | 630 | _ => e |

adam@347 | 631 end. |

adam@347 | 632 |

adam@347 | 633 Fixpoint optCmd (c : cmd) : cmd := |

adam@347 | 634 match c with |

adam@347 | 635 | Assign v e => Assign v (optExp e) |

adam@347 | 636 | Seq c1 c2 => Seq (optCmd c1) (optCmd c2) |

adam@347 | 637 | While e c => While (optExp e) (optCmd c) |

adam@347 | 638 end. |

adam@347 | 639 |

adam@347 | 640 (** Before proving correctness of [optCmd], we prove a lemma about [optExp]. This is where we have to do the most work, choosing pattern match opportunities automatically. *) |

adam@347 | 641 |

adam@347 | 642 (* begin thide *) |

adam@347 | 643 Lemma optExp_correct : forall vs e, evalExp vs (optExp e) = evalExp vs e. |

adam@347 | 644 induction e; crush; |

adam@347 | 645 repeat (match goal with |

adam@478 | 646 | [ |- context[match ?E with Const _ => _ | _ => _ end] ] => destruct E |

adam@347 | 647 | [ |- context[match ?E with O => _ | S _ => _ end] ] => destruct E |

adam@347 | 648 end; crush). |

adamc@64 | 649 Qed. |

adamc@64 | 650 |

adam@478 | 651 Hint Rewrite optExp_correct. |

adamc@64 | 652 |

adam@384 | 653 (** The final theorem is easy to establish, using our co-induction principle and a bit of Ltac smarts that we leave unexplained for now. Curious readers can consult the Coq manual, or wait for the later chapters of this book about proof automation. At a high level, we show inclusions between behaviors, going in both directions between original and optimized programs. *) |

adamc@64 | 654 |

adam@384 | 655 Ltac finisher := match goal with |

adam@384 | 656 | [ H : evalCmd _ _ _ |- _ ] => ((inversion H; []) |

adam@384 | 657 || (inversion H; [|])); subst |

adam@384 | 658 end; crush; eauto 10. |

adam@384 | 659 |

adam@384 | 660 Lemma optCmd_correct1 : forall vs1 c vs2, evalCmd vs1 c vs2 |

adam@347 | 661 -> evalCmd vs1 (optCmd c) vs2. |

adam@347 | 662 intros; apply (evalCmd_coind (fun vs1 c' vs2 => exists c, evalCmd vs1 c vs2 |

adam@347 | 663 /\ c' = optCmd c)); eauto; crush; |

adam@347 | 664 match goal with |

adam@347 | 665 | [ H : _ = optCmd ?E |- _ ] => destruct E; simpl in *; discriminate |

adam@347 | 666 || injection H; intros; subst |

adam@384 | 667 end; finisher. |

adam@384 | 668 Qed. |

adam@384 | 669 |

adam@384 | 670 Lemma optCmd_correct2 : forall vs1 c vs2, evalCmd vs1 (optCmd c) vs2 |

adam@384 | 671 -> evalCmd vs1 c vs2. |

adam@384 | 672 intros; apply (evalCmd_coind (fun vs1 c vs2 => evalCmd vs1 (optCmd c) vs2)); |

adam@384 | 673 crush; finisher. |

adam@384 | 674 Qed. |

adam@384 | 675 |

adam@384 | 676 Theorem optCmd_correct : forall vs1 c vs2, evalCmd vs1 (optCmd c) vs2 |

adam@384 | 677 <-> evalCmd vs1 c vs2. |

adam@384 | 678 intuition; apply optCmd_correct1 || apply optCmd_correct2; assumption. |

adam@347 | 679 Qed. |

adam@347 | 680 (* end thide *) |

adamc@64 | 681 |

adam@422 | 682 (** In this form, the theorem tells us that the optimizer preserves observable behavior of both terminating and nonterminating programs, but we did not have to do more work than for the case of terminating programs alone. We merely took the natural inductive definition for terminating executions, made it co-inductive, and applied the appropriate co-induction principle. Curious readers might experiment with adding command constructs like <<if>>; the same proof script should continue working, after the co-induction principle is extended to the new evaluation rules. *) |