### changeset 212:3227be370687

Ported Subset
author Adam Chlipala Mon, 09 Nov 2009 15:43:56 -0500 d06726f49bc6 c4b1c0de7af9 Makefile src/Subset.v 2 files changed, 75 insertions(+), 68 deletions(-) [+]
line wrap: on
line diff
--- a/Makefile	Mon Nov 09 14:48:46 2009 -0500
+++ b/Makefile	Mon Nov 09 15:43:56 2009 -0500
@@ -38,7 +38,9 @@
-o ../latex/cpdt.tex

latex/%.tex: src/%.v
-	coqdoc --interpolate --latex -s $< -o$@
+	coqdoc --interpolate --latex -s \
+		-p "\usepackage{url}" \
+		$< -o$@

latex/%.dvi: latex/%.tex
cd latex ; latex $* ; latex$*
--- a/src/Subset.v	Mon Nov 09 14:48:46 2009 -0500
+++ b/src/Subset.v	Mon Nov 09 15:43:56 2009 -0500
@@ -28,16 +28,16 @@
(** Let us consider several ways of implementing the natural number predecessor function.  We start by displaying the definition from the standard library: *)

Print pred.
-(** [[
-
+(** %\vspace{-.15in}% [[
pred = fun n : nat => match n with
| 0 => 0
| S u => u
end
: nat -> nat
-]] *)
+
+]]

-(** We can use a new command, [Extraction], to produce an OCaml version of this function. *)
+We can use a new command, [Extraction], to produce an OCaml version of this function. *)

Extraction pred.

@@ -64,35 +64,35 @@
Qed.

Definition pred_strong1 (n : nat) : n > 0 -> nat :=
-  match n return (n > 0 -> nat) with
+  match n with
| O => fun pf : 0 > 0 => match zgtz pf with end
| S n' => fun _ => n'
end.

(** We expand the type of [pred] to include a %\textit{%#<i>#proof#</i>#%}% that its argument [n] is greater than 0.  When [n] is 0, we use the proof to derive a contradiction, which we can use to build a value of any type via a vacuous pattern match.  When [n] is a successor, we have no need for the proof and just return the answer.  The proof argument can be said to have a %\textit{%#<i>#dependent#</i>#%}% type, because its type depends on the %\textit{%#<i>#value#</i>#%}% of the argument [n].

-There are two aspects of the definition of [pred_strong1] that may be surprising.  First, we took advantage of [Definition]'s syntactic sugar for defining function arguments in the case of [n], but we bound the proofs later with explicit [fun] expressions.  Second, there is the [return] clause for the [match], which we saw briefly in Chapter 2.  Let us see what happens if we write this function in the way that at first seems most natural. *)
+One aspects in particular of the definition of [pred_strong1] that may be surprising.  We took advantage of [Definition]'s syntactic sugar for defining function arguments in the case of [n], but we bound the proofs later with explicit [fun] expressions.  Let us see what happens if we write this function in the way that at first seems most natural.

-(** [[
+[[
Definition pred_strong1' (n : nat) (pf : n > 0) : nat :=
match n with
| O => match zgtz pf with end
| S n' => n'
end.

-  [[
Error: In environment
n : nat
pf : n > 0
The term "pf" has type "n > 0" while it is expected to have type
"0 > 0"
+
]]

-The term [zgtz pf] fails to type-check.  Somehow the type checker has failed to take into account information that follows from which [match] branch that term appears in.  The problem is that, by default, [match] does not let us use such implied information.  To get refined typing, we must always add special [match] annotations.
+The term [zgtz pf] fails to type-check.  Somehow the type checker has failed to take into account information that follows from which [match] branch that term appears in.  The problem is that, by default, [match] does not let us use such implied information.  To get refined typing, we must always rely on [match] annotations, either written explicitly or inferred.

In this case, we must use a [return] annotation to declare the relationship between the %\textit{%#<i>#value#</i>#%}% of the [match] discriminee and the %\textit{%#<i>#type#</i>#%}% of the result.  There is no annotation that lets us declare a relationship between the discriminee and the type of a variable that is already in scope; hence, we delay the binding of [pf], so that we can use the [return] annotation to express the needed relationship.

-Why does Coq not infer this relationship for us?  Certainly, it is not hard to imagine heuristics that would handle this particular case and many others.  In general, however, the inference problem is undecidable.  The known undecidable problem of %\textit{%#<i>#higher-order unification#</i>#%}% reduces to the [match] type inference problem.  Over time, Coq is enhanced with more and more heuristics to get around this problem, but there must always exist [match]es whose types Coq cannot infer without annotations.
+We are lucky that Coq's heuristics infer the [return] clause (specifically, [return n > 0 -> nat]) for us in this case.  In general, however, the inference problem is undecidable.  The known undecidable problem of %\textit{%#<i>#higher-order unification#</i>#%}% reduces to the [match] type inference problem.  Over time, Coq is enhanced with more and more heuristics to get around this problem, but there must always exist [match]es whose types Coq cannot infer without annotations.

Let us now take a look at the OCaml code Coq generates for [pred_strong1]. *)

@@ -119,12 +119,12 @@
We can reimplement our dependently-typed [pred] based on %\textit{%#<i>#subset types#</i>#%}%, defined in the standard library with the type family [sig]. *)

Print sig.
-(** [[
-
+(** %\vspace{-.15in}% [[
Inductive sig (A : Type) (P : A -> Prop) : Type :=
exist : forall x : A, P x -> sig P
For sig: Argument A is implicit
For exist: Argument A is implicit
+
]]

[sig] is a Curry-Howard twin of [ex], except that [sig] is in [Type], while [ex] is in [Prop].  That means that [sig] values can survive extraction, while [ex] proofs will always be erased.  The actual details of extraction of [sig]s are more subtle, as we will see shortly.
@@ -132,8 +132,7 @@
We rewrite [pred_strong1], using some syntactic sugar for subset types. *)

Locate "{ _ : _ | _ }".
-(** [[
-
+(** %\vspace{-.15in}% [[
Notation            Scope
"{ x : A  |  P }" := sig (fun x : A => P)
: type_scope
@@ -171,10 +170,10 @@
Definition pred_strong3 (s : {n : nat | n > 0}) : {m : nat | proj1_sig s = S m} :=
match s return {m : nat | proj1_sig s = S m} with
| exist 0 pf => match zgtz pf with end
-    | exist (S n') _ => exist _ n' (refl_equal _)
+    | exist (S n') pf => exist _ n' (refl_equal _)
end.

-(** The function [proj1_sig] extracts the base value from a subset type.  Besides the use of that function, the only other new thing is the use of the [exist] constructor to build a new [sig] value, and the details of how to do that follow from the output of our earlier [Print] command.
+(** The function [proj1_sig] extracts the base value from a subset type.  Besides the use of that function, the only other new thing is the use of the [exist] constructor to build a new [sig] value, and the details of how to do that follow from the output of our earlier [Print] command.  It also turns out that we need to include an explicit [return] clause here, since Coq's heuristics are not smart enough to propagate the result type that we wrote earlier.

By now, the reader is probably ready to believe that the new [pred_strong] leads to the same OCaml code as we have seen several times so far, and Coq does not disappoint. *)

@@ -200,34 +199,32 @@

Definition pred_strong4 (n : nat) : n > 0 -> {m : nat | n = S m}.
refine (fun n =>
-    match n return (n > 0 -> {m : nat | n = S m}) with
+    match n with
| O => fun _ => False_rec _ _
| S n' => fun _ => exist _ n' _
end).
+
(* begin thide *)
(** We build [pred_strong4] using tactic-based proving, beginning with a [Definition] command that ends in a period before a definition is given.  Such a command enters the interactive proving mode, with the type given for the new identifier as our proof goal.  We do most of the work with the [refine] tactic, to which we pass a partial "proof" of the type we are trying to prove.  There may be some pieces left to fill in, indicated by underscores.  Any underscore that Coq cannot reconstruct with type inference is added as a proof subgoal.  In this case, we have two subgoals:

[[
-
2 subgoals

n : nat
_ : 0 > 0
============================
False
-]]
-
-[[

subgoal 2 is:
S n' = S n'
+
]]

We can see that the first subgoal comes from the second underscore passed to [False_rec], and the second subgoal comes from the second underscore passed to [exist].  In the first case, we see that, though we bound the proof variable with an underscore, it is still available in our proof context.  It is hard to refer to underscore-named variables in manual proofs, but automation makes short work of them.  Both subgoals are easy to discharge that way, so let us back up and ask to prove all subgoals automatically. *)

Undo.
refine (fun n =>
-    match n return (n > 0 -> {m : nat | n = S m}) with
+    match n with
| O => fun _ => False_rec _ _
| S n' => fun _ => exist _ n' _
end); crush.
@@ -237,8 +234,7 @@
(** We end the "proof" with [Defined] instead of [Qed], so that the definition we constructed remains visible.  This contrasts to the case of ending a proof with [Qed], where the details of the proof are hidden afterward.  Let us see what our proof script constructed. *)

Print pred_strong4.
-(** [[
-
+(** %\vspace{-.15in}% [[
pred_strong4 =
fun n : nat =>
match n as n0 return (n0 > 0 -> {m : nat | n0 = S m}) with
@@ -254,6 +250,7 @@
exist (fun m : nat => S n' = S m) n' (refl_equal (S n'))
end
: forall n : nat, n > 0 -> {m : nat | n = S m}
+
]]

We see the code we entered, with some proofs filled in.  The first proof obligation, the second argument to [False_rec], is filled in with a nasty-looking proof term that we can be glad we did not enter by hand.  The second proof obligation is a simple reflexivity proof.
@@ -265,27 +262,39 @@

Definition pred_strong5 (n : nat) : n > 0 -> {m : nat | n = S m}.
refine (fun n =>
-    match n return (n > 0 -> {m : nat | n = S m}) with
+    match n with
| O => fun _ => !
| S n' => fun _ => [n']
end); crush.
Defined.

+(** One other alternative is worth demonstrating.  Recent Coq versions include a facility called [Program] that streamlines this style of definition.  Here is a complete implementation using [Program]. *)
+
+Obligation Tactic := crush.
+
+Program Definition pred_strong6 (n : nat) (_ : n > 0) : {m : nat | n = S m} :=
+  match n with
+    | O => _
+    | S n' => n'
+  end.
+
+(** Printing the resulting definition of [pred_strong6] yields a term very similar to what we built with [refine].  [Program] can save time in writing programs that use subset types.  Nonetheless, [refine] is often just as effective, and [refine] gives you more control over the form the final term takes, which can be useful when you want to prove additional theorems about your definition.  [Program] will sometimes insert type casts that can complicate theorem-proving. *)
+

(** * Decidable Proposition Types *)

(** There is another type in the standard library which captures the idea of program values that indicate which of two propositions is true. *)

Print sumbool.
-(** [[
-
+(** %\vspace{-.15in}% [[
Inductive sumbool (A : Prop) (B : Prop) : Set :=
left : A -> {A} + {B} | right : B -> {A} + {B}
For left: Argument A is implicit
For right: Argument B is implicit
-]] *)
+
+]]

-(** We can define some notations to make working with [sumbool] more convenient. *)
+We can define some notations to make working with [sumbool] more convenient. *)

Notation "'Yes'" := (left _ _).
Notation "'No'" := (right _ _).
@@ -296,8 +305,8 @@
Now we can write [eq_nat_dec], which compares two natural numbers, returning either a proof of their equality or a proof of their inequality. *)

Definition eq_nat_dec (n m : nat) : {n = m} + {n <> m}.
-  refine (fix f (n m : nat) {struct n} : {n = m} + {n <> m} :=
-    match n, m return {n = m} + {n <> m} with
+  refine (fix f (n m : nat) : {n = m} + {n <> m} :=
+    match n, m with
| O, O => Yes
| S n', S m' => Reduce (f n' m')
| _, _ => No
@@ -386,10 +395,9 @@

(** The final function is easy to write using the techniques we have developed so far. *)

-  Definition In_dec : forall (x : A) (ls : list A), {In x ls} + { ~In x ls}.
-    refine (fix f (x : A) (ls : list A) {struct ls}
-      : {In x ls} + { ~In x ls} :=
-      match ls return {In x ls} + { ~In x ls} with
+  Definition In_dec : forall (x : A) (ls : list A), {In x ls} + {~ In x ls}.
+    refine (fix f (x : A) (ls : list A) : {In x ls} + {~ In x ls} :=
+      match ls with
| nil => No
| x' :: ls' => A_eq_dec x x' || f x ls'
end); crush.
@@ -440,19 +448,18 @@

(** Now our next version of [pred] is trivial to write. *)

-Definition pred_strong6 (n : nat) : {{m | n = S m}}.
+Definition pred_strong7 (n : nat) : {{m | n = S m}}.
refine (fun n =>
-    match n return {{m | n = S m}} with
+    match n with
| O => ??
| S n' => [[n']]
end); trivial.
Defined.

-(** Because we used [maybe], one valid implementation of the type we gave [pred_strong6] would return [??] in every case.  We can strengthen the type to rule out such vacuous implementations, and the type family [sumor] from the standard library provides the easiest starting point.  For type [A] and proposition [B], [A + {B}] desugars to [sumor A B], whose values are either values of [A] or proofs of [B]. *)
+(** Because we used [maybe], one valid implementation of the type we gave [pred_strong7] would return [??] in every case.  We can strengthen the type to rule out such vacuous implementations, and the type family [sumor] from the standard library provides the easiest starting point.  For type [A] and proposition [B], [A + {B}] desugars to [sumor A B], whose values are either values of [A] or proofs of [B]. *)

Print sumor.
-(** [[
-
+(** %\vspace{-.15in}% [[
Inductive sumor (A : Type) (B : Prop) : Type :=
inleft : A -> A + {B} | inright : B -> A + {B}
For inleft: Argument A is implicit
@@ -466,9 +473,9 @@

(** Now we are ready to give the final version of possibly-failing predecessor.  The [sumor]-based type that we use is maximally expressive; any implementation of the type has the same input-output behavior. *)

-Definition pred_strong7 (n : nat) : {m : nat | n = S m} + {n = 0}.
+Definition pred_strong8 (n : nat) : {m : nat | n = S m} + {n = 0}.
refine (fun n =>
-    match n return {m : nat | n = S m} + {n = 0} with
+    match n with
| O => !!
| S n' => [[[n']]]
end); trivial.
@@ -491,8 +498,8 @@

Definition doublePred (n1 n2 : nat) : {{p | n1 = S (fst p) /\ n2 = S (snd p)}}.
refine (fun n1 n2 =>
-    m1 <- pred_strong6 n1;
-    m2 <- pred_strong6 n2;
+    m1 <- pred_strong7 n1;
+    m2 <- pred_strong7 n2;
[[(m1, m2)]]); tauto.
Defined.

@@ -508,11 +515,12 @@

(** printing * $\times$ *)

-Definition doublePred' (n1 n2 : nat) : {p : nat * nat | n1 = S (fst p) /\ n2 = S (snd p)}
+Definition doublePred' (n1 n2 : nat)
+  : {p : nat * nat | n1 = S (fst p) /\ n2 = S (snd p)}
+ {n1 = 0 \/ n2 = 0}.
refine (fun n1 n2 =>
-    m1 <-- pred_strong7 n1;
-    m2 <-- pred_strong7 n2;
+    m1 <-- pred_strong8 n1;
+    m2 <-- pred_strong8 n2;
[[[(m1, m2)]]]); tauto.
Defined.

@@ -552,7 +560,7 @@
decide equality.
Defined.

-(** Another notation complements the monadic notation for [maybe] that we defined earlier.  Sometimes we want to be to include "assertions" in our procedures.  That is, we want to run a decision procedure and fail if it fails; otherwise, we want to continue, with the proof that it produced made available to us.  This infix notation captures that, for a procedure that returns an arbitrary two-constructor type. *)
+(** Another notation complements the monadic notation for [maybe] that we defined earlier.  Sometimes we want to include "assertions" in our procedures.  That is, we want to run a decision procedure and fail if it fails; otherwise, we want to continue, with the proof that it produced made available to us.  This infix notation captures that idea, for a procedure that returns an arbitrary two-constructor type. *)

Notation "e1 ;; e2" := (if e1 then e2 else ??)
(right associativity, at level 60).
@@ -565,7 +573,7 @@
Hint Constructors hasType.

refine (fix F (e : exp) : {{t | hasType e t}} :=
-    match e return {{t | hasType e t}} with
+    match e with
| Nat _ => [[TNat]]
| Plus e1 e2 =>
t1 <- F e1;
@@ -587,22 +595,19 @@
(** Despite manipulating proofs, our type checker is easy to run. *)

Eval simpl in typeCheck (Nat 0).
-(** [[
-
+(** %\vspace{-.15in}% [[
= [[TNat]]
: {{t | hasType (Nat 0) t}}
]] *)

Eval simpl in typeCheck (Plus (Nat 1) (Nat 2)).
-(** [[
-
+(** %\vspace{-.15in}% [[
= [[TNat]]
: {{t | hasType (Plus (Nat 1) (Nat 2)) t}}
]] *)

Eval simpl in typeCheck (Plus (Nat 1) (Bool false)).
-(** [[
-
+(** %\vspace{-.15in}% [[
= ??
: {{t | hasType (Plus (Nat 1) (Bool false)) t}}
]] *)
@@ -700,7 +705,7 @@
(** Now we can define the type-checker.  Its type expresses that it only fails on untypable expressions. *)

(* end thide *)
-Definition typeCheck' (e : exp) : {t : type | hasType e t} + {forall t, ~hasType e t}.
+Definition typeCheck' (e : exp) : {t : type | hasType e t} + {forall t, ~ hasType e t}.
(* begin thide *)
Hint Constructors hasType.
(** We register all of the typing rules as hints. *)
@@ -709,8 +714,9 @@
(** [hasType_det] will also be useful for proving proof obligations with contradictory contexts.  Since its statement includes [forall]-bound variables that do not appear in its conclusion, only [eauto] will apply this hint. *)

(** Finally, the implementation of [typeCheck] can be transcribed literally, simply switching notations as needed. *)
-  refine (fix F (e : exp) : {t : type | hasType e t} + {forall t, ~hasType e t} :=
-    match e return {t : type | hasType e t} + {forall t, ~hasType e t} with
+
+  refine (fix F (e : exp) : {t : type | hasType e t} + {forall t, ~ hasType e t} :=
+    match e with
| Nat _ => [[[TNat]]]
| Plus e1 e2 =>
t1 <-- F e1;
@@ -729,6 +735,8 @@

(** We clear [F], the local name for the recursive function, to avoid strange proofs that refer to recursive calls that we never make.  The [crush] variant [crush'] helps us by performing automatic inversion on instances of the predicates specified in its second argument.  Once we throw in [eauto] to apply [hasType_det] for us, we have discharged all the subgoals. *)
(* end thide *)
+
+
Defined.

(** The short implementation here hides just how time-saving automation is.  Every use of one of the notations adds a proof obligation, giving us 12 in total.  Most of these obligations require multiple inversions and either uses of [hasType_det] or applications of [hasType] rules.
@@ -736,24 +744,21 @@
The results of simplifying calls to [typeCheck'] look deceptively similar to the results for [typeCheck], but now the types of the results provide more information. *)

Eval simpl in typeCheck' (Nat 0).
-(** [[
-
+(** %\vspace{-.15in}% [[
= [[[TNat]]]
: {t : type | hasType (Nat 0) t} +
{(forall t : type, ~ hasType (Nat 0) t)}
]] *)

Eval simpl in typeCheck' (Plus (Nat 1) (Nat 2)).
-(** [[
-
+(** %\vspace{-.15in}% [[
= [[[TNat]]]
: {t : type | hasType (Plus (Nat 1) (Nat 2)) t} +
{(forall t : type, ~ hasType (Plus (Nat 1) (Nat 2)) t)}
]] *)

Eval simpl in typeCheck' (Plus (Nat 1) (Bool false)).
-(** [[
-
+(** %\vspace{-.15in}% [[
= !!
: {t : type | hasType (Plus (Nat 1) (Bool false)) t} +
{(forall t : type, ~ hasType (Plus (Nat 1) (Bool false)) t)}
@@ -772,11 +777,11 @@
%\item%#<li># Define an inductive type [prop] of propositional logic formulas, consisting of variables, negation, and binary conjunction and disjunction.#</li>#
%\item%#<li># Define a function [propDenote] from variable truth assignments and [prop]s to [Prop], based on the usual meanings of the connectives.  Represent truth assignments as functions from [var] to [bool].#</li>#
%\item%#<li># Define a function [bool_true_dec] that checks whether a boolean is true, with a maximally expressive dependent type.  That is, the function should have type [forall b, {b = true} + {b = true -> False}]. #</li>#
-  %\item%#<li># Define a function [decide] that determines whether a particular [prop] is true under a particular truth assignment.  That is, the function should have type [forall (truth : var -> bool) (p : prop), {propDenote truth p} + { ~propDenote truth p}].  This function is probably easiest to write in the usual tactical style, instead of programming with [refine].  [bool_true_dec] may come in handy as a hint.#</li>#
-  %\item%#<li># Define a function [negate] that returns a simplified version of the negation of a [prop].  That is, the function should have type [forall p : prop, {p' : prop | forall truth, propDenote truth p <-> ~propDenote truth p'}].  To simplify a variable, just negate it.  Simplify a negation by returning its argument.  Simplify conjunctions and disjunctions using De Morgan's laws, negating the arguments recursively and switching the kind of connective.  [decide] may be useful in some of the proof obligations, even if you do not use it in the computational part of [negate]'s definition.  Lemmas like [decide] allow us to compensate for the lack of a general Law of the Excluded Middle in CIC.#</li>#
+  %\item%#<li># Define a function [decide] that determines whether a particular [prop] is true under a particular truth assignment.  That is, the function should have type [forall (truth : var -> bool) (p : prop), {propDenote truth p} + {~ propDenote truth p}].  This function is probably easiest to write in the usual tactical style, instead of programming with [refine].  [bool_true_dec] may come in handy as a hint.#</li>#
+  %\item%#<li># Define a function [negate] that returns a simplified version of the negation of a [prop].  That is, the function should have type [forall p : prop, {p' : prop | forall truth, propDenote truth p <-> ~ propDenote truth p'}].  To simplify a variable, just negate it.  Simplify a negation by returning its argument.  Simplify conjunctions and disjunctions using De Morgan's laws, negating the arguments recursively and switching the kind of connective.  [decide] may be useful in some of the proof obligations, even if you do not use it in the computational part of [negate]'s definition.  Lemmas like [decide] allow us to compensate for the lack of a general Law of the Excluded Middle in CIC.#</li>#
#</ol>#%\end{enumerate}% #</li>#

-%\item%#<li># Implement the DPLL satisfiability decision procedure for boolean formulas in conjunctive normal form, with a dependent type that guarantees its correctness.  An example of a reasonable type for this function would be [forall f : formula, {truth : tvals | formulaTrue truth f} + {forall truth, ~formulaTrue truth f}].  Implement at least "the basic backtracking algorithm" as defined here:
+%\item%#<li># Implement the DPLL satisfiability decision procedure for boolean formulas in conjunctive normal form, with a dependent type that guarantees its correctness.  An example of a reasonable type for this function would be [forall f : formula, {truth : tvals | formulaTrue truth f} + {forall truth, ~ formulaTrue truth f}].  Implement at least "the basic backtracking algorithm" as defined here:
%\begin{center}\url{http://en.wikipedia.org/wiki/DPLL_algorithm}\end{center}%
#<blockquote><a href="http://en.wikipedia.org/wiki/DPLL_algorithm">http://en.wikipedia.org/wiki/DPLL_algorithm</a></blockquote>#
It might also be instructive to implement the unit propagation and pure literal elimination optimizations described there or some other optimizations that have been used in modern SAT solvers.#</li>#