## Mercurial > cpdt > repo

### diff src/DataStruct.v @ 480:f38a3af9dd17

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Batch of changes based on proofreader feedback

author | Adam Chlipala <adam@chlipala.net> |
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date | Fri, 30 Nov 2012 11:57:55 -0500 |

parents | f02b698aadb1 |

children | 31258618ef73 |

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--- a/src/DataStruct.v Wed Nov 28 19:33:21 2012 -0500 +++ b/src/DataStruct.v Fri Nov 30 11:57:55 2012 -0500 @@ -41,7 +41,7 @@ | First : forall n, fin (S n) | Next : forall n, fin n -> fin (S n). - (** An instance of [fin] is essentially a more richly typed copy of the natural numbers. Every element is a [First] iterated through applying [Next] a number of times that indicates which number is being selected. For instance, the three values of type [fin 3] are [First 2], [Next (First 1)], and [Next (Next (First 0))]. + (** An instance of [fin] is essentially a more richly typed copy of a prefix of the natural numbers. The type [fin n] is isomorphic to [{m : nat | m < n}]. Every element is a [First] iterated through applying [Next] a number of times that indicates which number is being selected. For instance, the three values of type [fin 3] are [First 2], [Next (First 1)], and [Next (Next (First 0))]. Now it is easy to pick a [Prop]-free type for a selection function. As usual, our first implementation attempt will not convince the type checker, and we will attack the deficiencies one at a time. [[ @@ -55,7 +55,7 @@ end end. ]] - %\vspace{-.15in}%We apply the usual wisdom of delaying arguments in [Fixpoint]s so that they may be included in [return] clauses. This still leaves us with a quandary in each of the [match] cases. First, we need to figure out how to take advantage of the contradiction in the [Nil] case. Every [fin] has a type of the form [S n], which cannot unify with the [O] value that we learn for [n] in the [Nil] case. The solution we adopt is another case of [match]-within-[return]. + %\vspace{-.15in}%We apply the usual wisdom of delaying arguments in [Fixpoint]s so that they may be included in [return] clauses. This still leaves us with a quandary in each of the [match] cases. First, we need to figure out how to take advantage of the contradiction in the [Nil] case. Every [fin] has a type of the form [S n], which cannot unify with the [O] value that we learn for [n] in the [Nil] case. The solution we adopt is another case of [match]-within-[return], with the [return] clause chosen carefully so that it returns the proper type [A] in case the [fin] index is [O], which we know is true here; and so that it returns an easy-to-inhabit type [unit] in the remaining, impossible cases, which nonetheless appear explicitly in the body of the [match]. [[ Fixpoint get n (ls : ilist n) : fin n -> A := match ls with @@ -200,7 +200,7 @@ | HNil : hlist nil | HCons : forall (x : A) (ls : list A), B x -> hlist ls -> hlist (x :: ls). - (** We can implement a variant of the last section's [get] function for [hlist]s. To get the dependent typing to work out, we will need to index our element selectors by the types of data that they point to.%\index{Gallina terms!member}% *) + (** We can implement a variant of the last section's [get] function for [hlist]s. To get the dependent typing to work out, we will need to index our element selectors (in type family [member]) by the types of data that they point to.%\index{Gallina terms!member}% *) (* end thide *) (* EX: Define an analogue to [get] for [hlist]s. *) @@ -248,7 +248,7 @@ Implicit Arguments HNext [A elm x ls]. (* end thide *) -(** By putting the parameters [A] and [B] in [Type], we allow some very higher-order uses. For instance, one use of [hlist] is for the simple heterogeneous lists that we referred to earlier. *) +(** By putting the parameters [A] and [B] in [Type], we enable fancier kinds of polymorphism than in mainstream functional languages. For instance, one use of [hlist] is for the simple heterogeneous lists that we referred to earlier. *) Definition someTypes : list Set := nat :: bool :: nil. @@ -370,7 +370,7 @@ (** * Recursive Type Definitions *) -(** %\index{recursive type definition}%There is another style of datatype definition that leads to much simpler definitions of the [get] and [hget] definitions above. Because Coq supports "type-level computation," we can redo our inductive definitions as _recursive_ definitions. *) +(** %\index{recursive type definition}%There is another style of datatype definition that leads to much simpler definitions of the [get] and [hget] definitions above. Because Coq supports "type-level computation," we can redo our inductive definitions as _recursive_ definitions. Here we will preface type names with the letter [f] to indicate that they are based on explicit recursive _function_ definitions. *) (* EX: Come up with an alternate [ilist] definition that makes it easier to write [get]. *)