Mercurial > cpdt > repo

comparison src/Coinductive.v @ 392:4b1242b277b2

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
Typo fixes
author Adam Chlipala <adam@chlipala.net>
date Fri, 20 Apr 2012 12:49:47 -0400
parents eb0fa506d04c
children 3c941750c347
comparison
equal deleted inserted replaced
391:fd3f1057685c 392:4b1242b277b2
360 360
361 (** 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. 361 (** 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.
362 362
363 %\medskip% 363 %\medskip%
364 364
365 Must we always be cautious with automation in proofs by co-induction? Induction seems to have dual versions 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}\emph{%#<i>#co-induction principles#</i>#%}%. Let us take that tack here, so that we can arrive at an [induction x; crush]-style proof for [ones_eq']. 365 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}\emph{%#<i>#co-induction principles#</i>#%}%. Let us take that tack here, so that we can arrive at an [induction x; crush]-style proof for [ones_eq'].
366 366
367 An induction principle is parameterized over a predicate characterizing what we mean to prove, %\emph{%#<i>#as a function of the inductive fact that we already know#</i>#%}%. Dually, a co-induction principle ought to be parameterized over a predicate characterizing what we mean to prove, %\emph{%#<i>#as a function of the arguments to the co-inductive predicate that we are trying to prove#</i>#%}%. 367 An induction principle is parameterized over a predicate characterizing what we mean to prove, %\emph{%#<i>#as a function of the inductive fact that we already know#</i>#%}%. Dually, a co-induction principle ought to be parameterized over a predicate characterizing what we mean to prove, %\emph{%#<i>#as a function of the arguments to the co-inductive predicate that we are trying to prove#</i>#%}%.
368 368
369 To state a useful principle for [stream_eq], it will be useful first to define the stream head function. *) 369 To state a useful principle for [stream_eq], it will be useful first to define the stream head function. *)
370 370
376 (** 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}%. *) 376 (** 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}%. *)
377 377
378 Section stream_eq_coind. 378 Section stream_eq_coind.
379 Variable A : Type. 379 Variable A : Type.
380 Variable R : stream A -> stream A -> Prop. 380 Variable R : stream A -> stream A -> Prop.
381 (** This relation generalizes the theorem we want to prove, characterizinge exactly which two arguments to [stream_eq] we want to consider. *) 381 (** This relation generalizes the theorem we want to prove, characterizing exactly which two arguments to [stream_eq] we want to consider. *)
382 382
383 Hypothesis Cons_case_hd : forall s1 s2, R s1 s2 -> hd s1 = hd s2. 383 Hypothesis Cons_case_hd : forall s1 s2, R s1 s2 -> hd s1 = hd s2.
384 Hypothesis Cons_case_tl : forall s1 s2, R s1 s2 -> R (tl s1) (tl s2). 384 Hypothesis Cons_case_tl : forall s1 s2, R s1 s2 -> R (tl s1) (tl s2).
385 (** 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 a [R] stream pair passes on %``%#"#[R]-ness#"#%''% to its tails. An established technical term for such a relation is %\index{bisimulation}\emph{%#<i>#bisimulation#</i>#%}%. *) 385 (** 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}\emph{%#<i>#bisimulation#</i>#%}%. *)
386 386
387 (** 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. *) 387 (** 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. *)
388
388 Theorem stream_eq_coind : forall s1 s2, R s1 s2 -> stream_eq s1 s2. 389 Theorem stream_eq_coind : forall s1 s2, R s1 s2 -> stream_eq s1 s2.
389 cofix; destruct s1; destruct s2; intro. 390 cofix; destruct s1; destruct s2; intro.
390 generalize (Cons_case_hd H); intro Heq; simpl in Heq; rewrite Heq. 391 generalize (Cons_case_hd H); intro Heq; simpl in Heq; rewrite Heq.
391 constructor. 392 constructor.
392 apply stream_eq_coind. 393 apply stream_eq_coind.
393 apply (Cons_case_tl H). 394 apply (Cons_case_tl H).
394 Qed. 395 Qed.
395 End stream_eq_coind. 396 End stream_eq_coind.
396 397
397 (** 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. *) 398 (** 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. *)
399
398 Print stream_eq_coind. 400 Print stream_eq_coind.
399 401
400 (** 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. *) 402 (** 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. *)
401 403
402 Theorem ones_eq'' : stream_eq ones ones'. 404 Theorem ones_eq'' : stream_eq ones ones'.