changeset 202:8caa3b3f8fc0

Feedback from Peter Gammie
author Adam Chlipala Sat, 03 Jan 2009 19:57:02 -0500 a2b14ba218a7 71ade09024ac src/Coinductive.v src/InductiveTypes.v src/Predicates.v src/StackMachine.v 4 files changed, 19 insertions(+), 22 deletions(-) [+]
line wrap: on
line diff
--- a/src/Coinductive.v	Fri Jan 02 09:09:35 2009 -0500
+++ b/src/Coinductive.v	Sat Jan 03 19:57:02 2009 -0500
@@ -22,7 +22,13 @@

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.

-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>#%}%.
+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>#%}%.  For an arbitrary proposition [P], we could write:
+
+[[
+
+
+This would leave us with [bad tt] as a proof of [P].

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.

@@ -83,15 +89,12 @@
: list bool
]] *)

-(* begin thide *)
(** 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.

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.

[[
-(* end thide *)
CoFixpoint looper : stream nat := looper.
-(* begin thide *)
[[
Error:
Recursive definition of looper is ill-formed.
@@ -100,7 +103,7 @@

unguarded recursive call in "looper"
*)
-(* end thide *)
+

(** 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.

@@ -137,13 +140,11 @@

(* begin thide *)
(** [[
-(* end thide *)

CoFixpoint map' (s : stream A) : stream B :=
match s with
| Cons h t => interleave (Cons (f h) (map' s)) (Cons (f h) (map' s))
end.
-(* begin thide *)
*)

(** 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.
@@ -359,7 +360,7 @@
end.
End regmap.

-(** Now comes the interesting part. We define a co-inductive semantics for programs.  The definition itself is not surprising.  We could change [CoInductive] to [Inductive] and arrive at a valid semantics that only covers terminating program executions.  Using [CoInductive] admits infinite derivations for infinite executions. *)
+(** Now comes the interesting part. We define a co-inductive semantics for programs.  The definition itself is not surprising.  We could change [CoInductive] to [Inductive] and arrive at a valid semantics that only covers terminating program executions.  Using [CoInductive] admits infinite derivations for infinite executions.  An application [run rm is v] means that, when we run the instructions [is] starting with register map [rm], either execution terminates with result [v] or execution runs safely forever.  (That is, the choice of [v] is immaterial for non-terminating executions.) *)

Section run.
Variable prog : program.
--- a/src/InductiveTypes.v	Fri Jan 02 09:09:35 2009 -0500
+++ b/src/InductiveTypes.v	Sat Jan 03 19:57:02 2009 -0500
@@ -449,19 +449,15 @@

(** After we end the section, the [Variable]s we used are added as extra function parameters for each defined identifier, as needed. *)

-Check list.
+Print list.
(** [[

-list
-     : Set -> Set
-]] *)

-Check Cons.
-(** [[
+Inductive list (T : Set) : Set :=
+    Nil : list T | Cons : T -> list T -> list Tlist
+]]

-Cons
-     : forall T : Set, T -> list T -> list T
-]] *)
+The final definition is the same as what we wrote manually before.  The other elements of the section are altered similarly, turning out exactly as they were before, though we managed to write their definitions more succinctly. *)

Check length.
(** [[
@@ -470,7 +466,7 @@
: forall T : Set, list T -> nat
]]

-The extra parameter [T] is treated as a new argument to the induction principle, too. *)
+The parameter [T] is treated as a new argument to the induction principle, too. *)

Check list_ind.
(** [[
@@ -1093,7 +1089,7 @@
f false
]]

-Now the righthand side of [H]'s equality appears in the conclusion, so we can rewrite. *)
+Now the righthand side of [H]'s equality appears in the conclusion, so we can rewrite, using the notation [<-] to request to replace the righthand side the equality with the lefthand side. *)

rewrite <- H.
(** [[
--- a/src/Predicates.v	Fri Jan 02 09:09:35 2009 -0500
+++ b/src/Predicates.v	Sat Jan 03 19:57:02 2009 -0500
@@ -443,7 +443,7 @@
Inductive eq (A : Type) (x : A) : A -> Prop :=  refl_equal : x = x
]] *)

-(** [eq] is the type we get behind the scenes when uses of infix [=] are expanded.  We see that [eq] has both a parameter [x] that is fixed and an extra unnamed argument of the same type.  It is crucial that the second argument is untied to the parameter in the type of [eq]; otherwise, we would have to prove that two values are equal even to be able to state the possibility of equality, which would more or less defeat the purpose of having an equality proposition.  However, examining the type of equality's sole constructor [refl_equal], we see that we can only %\textit{%#<i>#prove#</i>#%}% equality when its two arguments are syntactically equal.  This definition turns out to capture all of the basic properties of equality, and the equality-manipulating tactics that we have seen so far, like [reflexivity] and [rewrite], are implemented treating [eq] as just another inductive type with a well-chosen definition.
+(** [eq] is the type we get behind the scenes when uses of infix [=] are expanded.  We see that [eq] has both a parameter [x] that is fixed and an extra unnamed argument of the same type.  The type of [eq] allows us to state any equalities, even those that are provably false.  However, examining the type of equality's sole constructor [refl_equal], we see that we can only %\textit{%#<i>#prove#</i>#%}% equality when its two arguments are syntactically equal.  This definition turns out to capture all of the basic properties of equality, and the equality-manipulating tactics that we have seen so far, like [reflexivity] and [rewrite], are implemented treating [eq] as just another inductive type with a well-chosen definition.

Returning to the example of [isZero], we can see how to make use of hypotheses that use this predicate. *)

@@ -743,7 +743,7 @@
(** The induction hypothesis lets us complete the proof. *)
apply IHeven with n0; assumption.

-  (** As usual, we can rewrite the proof to avoid referencing any locally-generated names, which makes our proof script more readable and more robust to changes in the theorem statement. *)
+  (** As usual, we can rewrite the proof to avoid referencing any locally-generated names, which makes our proof script more readable and more robust to changes in the theorem statement.  We use the notation [<-] to request a hint that does right-to-left rewriting, just like we can with the [rewrite] tactic. *)
Restart.
Hint Rewrite <- plus_n_Sm : cpdt.

--- a/src/StackMachine.v	Fri Jan 02 09:09:35 2009 -0500
+++ b/src/StackMachine.v	Sat Jan 03 19:57:02 2009 -0500
@@ -26,7 +26,7 @@
There are some minor headaches associated with getting Proof General to pass the proper command line arguments to the %\texttt{%#<tt>#coqtop#</tt>#%}% program.  The best way to add settings that will be shared by many source files is to add a custom variable setting to your %\texttt{%#<tt>#.emacs#</tt>#%}% file, like this:
%\begin{verbatim}%#<pre>#(custom-set-variables
...
-  '(coq-prog-args '("-I" "/path/to/cpdt/src"))
+  '(coq-prog-args '("-impredicative-set" "-I" "/path/to/cpdt/src"))
...
)#</pre>#%\end{verbatim}%
The extra arguments demonstrated here are the proper choices for working with the code for this book.  The ellipses stand for other Emacs customization settings you may already have.  It can be helpful to save several alternate sets of flags in your %\texttt{%#<tt>#.emacs#</tt>#%}% file, with all but one commented out within the %\texttt{%#<tt>#custom-set-variables#</tt>#%}% block at any given time.