Undo some overzealous vspace tweaks

src/Coinductive.v

@@ -29,7 +29,7 @@ We spent some time in the last chapter discussing the %\index{Curry-Howard corre
 Fixpoint bad (u : unit) : P := bad u.
 ]]
 
-%\smallskip{}%This would leave us with [bad tt] as a proof of [P].
+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.
 

src/Predicates.v

@@ -45,7 +45,7 @@ Print True.
   Inductive True : Prop :=  I : True
   ]]
 
-%\smallskip{}%Recall that [unit] is the type with only one value, and [True] is the proposition that always holds.  Despite this superficial difference between the two concepts, in both cases we can use the same inductive definition mechanism.  The connection goes further than this.  We see that we arrive at the definition of [True] by replacing [unit] by [True], [tt] by [I], and [Set] by [Prop].  The first two of these differences are superficial changes of names, while the third difference is the crucial one for separating programs from proofs.  A term [T] of type [Set] is a type of programs, and a term of type [T] is a program.  A term [T] of type [Prop] is a logical proposition, and its proofs are of type [T].  Chapter 12 goes into more detail about the theoretical differences between [Prop] and [Set].  For now, we will simply follow common intuitions about what a proof is.
+Recall that [unit] is the type with only one value, and [True] is the proposition that always holds.  Despite this superficial difference between the two concepts, in both cases we can use the same inductive definition mechanism.  The connection goes further than this.  We see that we arrive at the definition of [True] by replacing [unit] by [True], [tt] by [I], and [Set] by [Prop].  The first two of these differences are superficial changes of names, while the third difference is the crucial one for separating programs from proofs.  A term [T] of type [Set] is a type of programs, and a term of type [T] is a program.  A term [T] of type [Prop] is a logical proposition, and its proofs are of type [T].  Chapter 12 goes into more detail about the theoretical differences between [Prop] and [Set].  For now, we will simply follow common intuitions about what a proof is.
 
 The type [unit] has one value, [tt].  The type [True] has one proof, [I].  Why distinguish between these two types?  Many people who have read about Curry-Howard in an abstract context and not put it to use in proof engineering answer that the two types in fact _should not_ be distinguished.  There is a certain aesthetic appeal to this point of view, but I want to argue that it is best to treat Curry-Howard very loosely in practical proving.  There are Coq-specific reasons for preferring the distinction, involving efficient compilation and avoidance of paradoxes in the presence of classical math, but I will argue that there is a more general principle that should lead us to avoid conflating programming and proving.
 
@@ -87,7 +87,7 @@ We have also already seen the definition of [True].  For a demonstration of a lo
   Inductive False : Prop :=
   ]]
 
-  %\smallskip{}%We can conclude anything from [False], doing case analysis on a proof of [False] in the same way we might do case analysis on, say, a natural number.  Since there are no cases to consider, any such case analysis succeeds immediately in proving the goal. *)
+  We can conclude anything from [False], doing case analysis on a proof of [False] in the same way we might do case analysis on, say, a natural number.  Since there are no cases to consider, any such case analysis succeeds immediately in proving the goal. *)
 
   Theorem False_imp : False -> 2 + 2 = 5.
 (* begin thide *)
@@ -129,7 +129,7 @@ We have also already seen the definition of [True].  For a demonstration of a lo
       : Prop -> Prop
      ]]
 
-     %\smallskip{}%We see that [not] is just shorthand for implication of [False].  We can use that fact explicitly in proofs.  The syntax [~ P] expands to [not P]. *)
+     We see that [not] is just shorthand for implication of [False].  We can use that fact explicitly in proofs.  The syntax [~ P] expands to [not P]. *)
 
   Theorem arith_neq' : ~ (2 + 2 = 5).
 (* begin thide *)
@@ -151,7 +151,7 @@ We have also already seen the definition of [True].  For a demonstration of a lo
     Inductive and (A : Prop) (B : Prop) : Prop :=  conj : A -> B -> A /\ B
   ]]
   
-  %\smallskip{}%The interested reader can check that [and] has a Curry-Howard equivalent called %\index{Gallina terms!prod}%[prod], the type of pairs.  However, it is generally most convenient to reason about conjunction using tactics.  An explicit proof of commutativity of [and] illustrates the usual suspects for such tasks.  The operator [/\] is an infix shorthand for [and]. *)
+  The interested reader can check that [and] has a Curry-Howard equivalent called %\index{Gallina terms!prod}%[prod], the type of pairs.  However, it is generally most convenient to reason about conjunction using tactics.  An explicit proof of commutativity of [and] illustrates the usual suspects for such tasks.  The operator [/\] is an infix shorthand for [and]. *)
 
   Theorem and_comm : P /\ Q -> Q /\ P.
 
@@ -199,7 +199,7 @@ subgoal 2 is
     or_introl : A -> A \/ B | or_intror : B -> A \/ B
 ]]
 
-%\smallskip{}%We see that there are two ways to prov a disjunction: prove the first disjunct or prove the second.  The Curry-Howard analogue of this is the Coq %\index{Gallina terms!sum}%[sum] type.  We can demonstrate the main tactics here with another proof of commutativity. *)
+We see that there are two ways to prov a disjunction: prove the first disjunct or prove the second.  The Curry-Howard analogue of this is the Coq %\index{Gallina terms!sum}%[sum] type.  We can demonstrate the main tactics here with another proof of commutativity. *)
 
   Theorem or_comm : P \/ Q -> Q \/ P.
 
@@ -373,7 +373,7 @@ We will see more about Coq's program extraction facility in a later chapter.  Ho
     ex_intro : forall x : A, P x -> ex P
     ]]
 
-  %\smallskip{}%The family [ex] is parameterized by the type [A] that we quantify over, and by a predicate [P] over [A]s.  We prove an existential by exhibiting some [x] of type [A], along with a proof of [P x].  As usual, there are tactics that save us from worrying about the low-level details most of the time.  We use the equality operator [=], which, depending on the settings in which they learned logic, different people will say either is or is not part of first-order logic.  For our purposes, it is. *)
+  The family [ex] is parameterized by the type [A] that we quantify over, and by a predicate [P] over [A]s.  We prove an existential by exhibiting some [x] of type [A], along with a proof of [P x].  As usual, there are tactics that save us from worrying about the low-level details most of the time.  We use the equality operator [=], which, depending on the settings in which they learned logic, different people will say either is or is not part of first-order logic.  For our purposes, it is. *)
 
 Theorem exist1 : exists x : nat, x + 1 = 2.
 (* begin thide *)
@@ -463,7 +463,7 @@ Print eq.
   Inductive eq (A : Type) (x : A) : A -> Prop :=  eq_refl : x = x
   ]]
 
-  %\smallskip{}%Behind the scenes, uses of infix [=] are expanded to instances of [eq].  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 [eq_refl], we see that we can only _prove_ 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.  Another way of stating that definition is: equality is defined as the least reflexive relation.
+  Behind the scenes, uses of infix [=] are expanded to instances of [eq].  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 [eq_refl], we see that we can only _prove_ 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.  Another way of stating that definition is: equality is defined as the least reflexive relation.
 
 Returning to the example of [isZero], we can see how to work with hypotheses that use this predicate. *)
 
@@ -531,7 +531,7 @@ isZero_ind
      : forall P : nat -> Prop, P 0 -> forall n : nat, isZero n -> P n
    ]]
 
-   %\smallskip{}%In our last proof script, [destruct] chose to instantiate [P] as [fun n => S n + S n = S (S (S (S n)))].  You can verify for yourself that this specialization of the principle applies to the goal and that the hypothesis [P 0] then matches the subgoal we saw generated.  If you are doing a proof and encounter a strange transmutation like this, there is a good chance that you should go back and replace a use of [destruct] with [inversion]. *)
+   In our last proof script, [destruct] chose to instantiate [P] as [fun n => S n + S n = S (S (S (S n)))].  You can verify for yourself that this specialization of the principle applies to the goal and that the hypothesis [P 0] then matches the subgoal we saw generated.  If you are doing a proof and encounter a strange transmutation like this, there is a good chance that you should go back and replace a use of [destruct] with [inversion]. *)
 
 
 (* begin hide *)

src/StackMachine.v

@@ -103,6 +103,9 @@ Eval simpl in expDenote (Binop Plus (Const 2) (Const 2)).
 Eval simpl in expDenote (Binop Times (Binop Plus (Const 2) (Const 2)) (Const 7)).
 (** [= 28 : nat] *)
 
+(** %\smallskip{}%Nothing too surprising goes on here, so we are ready to move on to the target language of our compiler. *)
+
+
 (** ** Target Language *)
 
 (** We will compile our source programs onto a simple stack machine, whose syntax is: *)
@@ -140,6 +143,8 @@ Fixpoint progDenote (p : prog) (s : stack) : option stack :=
       end
   end.
 
+(** With the two programming languages defined, we can turn to the compiler definition. *)
+
 
 (** ** Translation *)
 
@@ -163,7 +168,7 @@ Eval simpl in compile (Binop Plus (Const 2) (Const 2)).
 Eval simpl in compile (Binop Times (Binop Plus (Const 2) (Const 2)) (Const 7)).
 (** [= iConst 7 :: iConst 2 :: iConst 2 :: iBinop Plus :: iBinop Times :: nil : prog] *)
 
-(** We can also run our compiled programs and check that they give the right results. *)
+(** %\smallskip{}%We can also run our compiled programs and check that they give the right results. *)
 
 Eval simpl in progDenote (compile (Const 42)) nil.
 (** [= Some (42 :: nil) : option stack] *)
@@ -615,10 +620,12 @@ Eval simpl in texpDenote (TBinop TLt (TBinop TPlus (TNConst 2) (TNConst 2))
   (TNConst 7)).
 (** [= true : typeDenote Bool] *)
 
+(** %\smallskip{}%Now we are ready to define a suitable stack machine target for compilation. *)
+
 
 (** ** Target Language *)
 
-(** Now we want to define a suitable stack machine target for compilation.  In the example of the untyped language, stack machine programs could encounter stack underflows and "get stuck."  This was unfortunate, since we had to deal with this complication even though we proved that our compiler never produced underflowing programs.  We could have used dependent types to force all stack machine programs to be underflow-free.
+(** In the example of the untyped language, stack machine programs could encounter stack underflows and "get stuck."  This was unfortunate, since we had to deal with this complication even though we proved that our compiler never produced underflowing programs.  We could have used dependent types to force all stack machine programs to be underflow-free.
 
 For our new languages, besides underflow, we also have the problem of stack slots with naturals instead of bools or vice versa.  This time, we will use indexed typed families to avoid the need to reason about potential failures.
 
@@ -761,6 +768,8 @@ Eval simpl in tprogDenote (tcompile (TBinop TLt (TBinop TPlus (TNConst 2) (TNCon
   (TNConst 7)) nil) tt.
 (** [= (true, tt) : vstack (Bool :: nil)] *)
 
+(** %\smallskip{}%The compiler seems to be working, so let us turn to proving that it _always_ works. *)
+
 
 (** ** Translation Correctness *)
 

src/Subset.v

@@ -85,7 +85,7 @@ Eval compute in pred_strong1 two_gt0.
      : nat
  ]]
 
-%\smallskip{}%One aspect in particular of the definition of [pred_strong1] 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.
+One aspect in particular of the definition of [pred_strong1] 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.
 
 %\vspace{-.15in}%[[
 Definition pred_strong1' (n : nat) (pf : n > 0) : nat :=
@@ -153,7 +153,7 @@ Inductive sig (A : Type) (P : A -> Prop) : Type :=
     exist : forall x : A, P x -> sig P
 ]]
 
-%\smallskip{}%The family [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.
+The family [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.
 
 We rewrite [pred_strong1], using some syntactic sugar for subset types. *)
 
@@ -301,7 +301,7 @@ end
      : forall n : nat, n > 0 -> {m : nat | n = S m}
 ]]
 
-%\smallskip{}%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. *)
+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. *)
 
 Eval compute in pred_strong4 two_gt0.
 (** %\vspace{-.15in}% [[
@@ -309,7 +309,7 @@ Eval compute in pred_strong4 two_gt0.
      : {m : nat | 2 = S m}
      ]]
 
-  %\smallskip{}%A tactic modifier called %\index{tactics!abstract}%[abstract] can be helpful for producing shorter terms, by automatically abstracting subgoals into named lemmas. *)
+  A tactic modifier called %\index{tactics!abstract}%[abstract] can be helpful for producing shorter terms, by automatically abstracting subgoals into named lemmas. *)
 
 (* begin thide *)
 Definition pred_strong4' : forall n : nat, n > 0 -> {m : nat | n = S m}.
@@ -358,7 +358,7 @@ Eval compute in pred_strong5 two_gt0.
      : {m : nat | 2 = S m}
      ]]
 
-  %\smallskip{}%One other alternative is worth demonstrating.  Recent Coq versions include a facility called %\index{Program}%[Program] that streamlines this style of definition.  Here is a complete implementation using [Program].%\index{Vernacular commands!Obligation Tactic}\index{Vernacular commands!Program Definition}% *)
+  One other alternative is worth demonstrating.  Recent Coq versions include a facility called %\index{Program}%[Program] that streamlines this style of definition.  Here is a complete implementation using [Program].%\index{Vernacular commands!Obligation Tactic}\index{Vernacular commands!Program Definition}% *)
 
 Obligation Tactic := crush.
 
@@ -376,7 +376,7 @@ Eval compute in pred_strong6 two_gt0.
      : {m : nat | 2 = S m}
      ]]
 
-%\smallskip{}%In this case, we see that the new definition yields the same computational behavior as before. *)
+In this case, we see that the new definition yields the same computational behavior as before. *)
 
 
 (** * Decidable Proposition Types *)
@@ -421,7 +421,7 @@ Eval compute in eq_nat_dec 2 3.
      : {2 = 3} + {2 <> 3}
      ]]
 
-%\smallskip{}%Note that the %\coqdocnotation{%#<tt>#Yes#</tt>#%}% and %\coqdocnotation{%#<tt>#No#</tt>#%}% notations are hiding proofs establishing the correctness of the outputs.
+Note that the %\coqdocnotation{%#<tt>#Yes#</tt>#%}% and %\coqdocnotation{%#<tt>#No#</tt>#%}% notations are hiding proofs establishing the correctness of the outputs.
 
    Our definition extracts to reasonable OCaml code. *)
 
@@ -529,7 +529,7 @@ Eval compute in In_dec eq_nat_dec 3 (1 :: 2 :: nil).
      : {In 3 (1 :: 2 :: nil)} + {~ In 3 (1 :: 2 :: nil)}
      ]]
 
-%\smallskip{}%[In_dec] has a reasonable extraction to OCaml. *)
+[In_dec] has a reasonable extraction to OCaml. *)
 
 Extraction In_dec.
 (* end thide *)
@@ -596,7 +596,7 @@ Eval compute in pred_strong7 0.
      : {{m | 0 = S m}}
      ]]
 
-     %\smallskip{}%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 %\index{Gallina terms!sumor}%[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 %\index{Gallina terms!sumor}%[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}% [[
@@ -604,7 +604,7 @@ Inductive sumor (A : Type) (B : Prop) : Type :=
     inleft : A -> A + {B} | inright : B -> A + {B}
 ]]
 
-%\smallskip{}%We add notations for easy use of the [sumor] constructors.  The second notation is specialized to [sumor]s whose [A] parameters are instantiated with regular subset types, since this is how we will use [sumor] below. *)
+We add notations for easy use of the [sumor] constructors.  The second notation is specialized to [sumor]s whose [A] parameters are instantiated with regular subset types, since this is how we will use [sumor] below. *)
 
 Notation "!!" := (inright _ _).
 Notation "[|| x ||]" := (inleft _ [x]).
@@ -776,7 +776,7 @@ Eval simpl in typeCheck (Plus (Nat 1) (Bool false)).
      : {{t | hasType (Plus (Nat 1) (Bool false)) t}}
      ]]
 
-%\smallskip{}%The type checker also extracts to some reasonable OCaml code. *)
+The type checker also extracts to some reasonable OCaml code. *)
 
 Extraction typeCheck.
 
@@ -929,4 +929,4 @@ Eval simpl in typeCheck' (Plus (Nat 1) (Bool false)).
        {(forall t : type, ~ hasType (Plus (Nat 1) (Bool false)) t)}
        ]]
 
-%\smallskip{}%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. *)
+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. *)