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Pass through InductiveTypes, to incorporate new coqdoc features
author Adam Chlipala <adam@chlipala.net>
date Wed, 25 Jul 2012 16:15:50 -0400
parents 539ed97750bb
children 5f25705a10ea
rev   line source
adam@398 1 (* Copyright (c) 2008-2012, Adam Chlipala
adamc@105 2 *
adamc@105 3 * This work is licensed under a
adamc@105 4 * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
adamc@105 5 * Unported License.
adamc@105 6 * The license text is available at:
adamc@105 7 * http://creativecommons.org/licenses/by-nc-nd/3.0/
adamc@105 8 *)
adamc@105 9
adamc@105 10 (* begin hide *)
adamc@111 11 Require Import Arith List.
adamc@105 12
adam@314 13 Require Import CpdtTactics.
adamc@105 14
adamc@105 15 Set Implicit Arguments.
adamc@105 16 (* end hide *)
adamc@105 17
adamc@105 18
adamc@105 19 (** %\chapter{Dependent Data Structures}% *)
adamc@105 20
adamc@106 21 (** Our red-black tree example from the last chapter illustrated how dependent types enable static enforcement of data structure invariants. To find interesting uses of dependent data structures, however, we need not look to the favorite examples of data structures and algorithms textbooks. More basic examples like length-indexed and heterogeneous lists come up again and again as the building blocks of dependent programs. There is a surprisingly large design space for this class of data structure, and we will spend this chapter exploring it. *)
adamc@105 22
adamc@105 23
adamc@106 24 (** * More Length-Indexed Lists *)
adamc@106 25
adam@342 26 (** We begin with a deeper look at the length-indexed lists that began the last chapter.%\index{Gallina terms!ilist}% *)
adamc@105 27
adamc@105 28 Section ilist.
adamc@105 29 Variable A : Set.
adamc@105 30
adamc@105 31 Inductive ilist : nat -> Set :=
adamc@105 32 | Nil : ilist O
adamc@105 33 | Cons : forall n, A -> ilist n -> ilist (S n).
adamc@105 34
adam@342 35 (** We might like to have a certified function for selecting an element of an [ilist] by position. We could do this using subset types and explicit manipulation of proofs, but dependent types let us do it more directly. It is helpful to define a type family %\index{Gallina terms!fin}%[fin], where [fin n] is isomorphic to [{m : nat | m < n}]. The type family name stands for %``%#"#finite.#"#%''% *)
adamc@106 36
adamc@113 37 (* EX: Define a function [get] for extracting an [ilist] element by position. *)
adamc@113 38
adamc@113 39 (* begin thide *)
adamc@215 40 Inductive fin : nat -> Set :=
adamc@215 41 | First : forall n, fin (S n)
adamc@215 42 | Next : forall n, fin n -> fin (S n).
adamc@105 43
adam@406 44 (** 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))].
adamc@106 45
adamc@106 46 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.
adamc@106 47
adamc@106 48 [[
adamc@215 49 Fixpoint get n (ls : ilist n) : fin n -> A :=
adamc@215 50 match ls with
adamc@106 51 | Nil => fun idx => ?
adamc@106 52 | Cons _ x ls' => fun idx =>
adamc@106 53 match idx with
adamc@106 54 | First _ => x
adamc@106 55 | Next _ idx' => get ls' idx'
adamc@106 56 end
adamc@106 57 end.
adamc@106 58
adamc@205 59 ]]
adamc@205 60
adamc@215 61 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].
adamc@106 62
adamc@106 63 [[
adamc@215 64 Fixpoint get n (ls : ilist n) : fin n -> A :=
adamc@215 65 match ls with
adamc@106 66 | Nil => fun idx =>
adamc@215 67 match idx in fin n' return (match n' with
adamc@106 68 | O => A
adamc@106 69 | S _ => unit
adamc@106 70 end) with
adamc@106 71 | First _ => tt
adamc@106 72 | Next _ _ => tt
adamc@106 73 end
adamc@106 74 | Cons _ x ls' => fun idx =>
adamc@106 75 match idx with
adamc@106 76 | First _ => x
adamc@106 77 | Next _ idx' => get ls' idx'
adamc@106 78 end
adamc@106 79 end.
adamc@106 80
adamc@205 81 ]]
adamc@205 82
adam@284 83 Now the first [match] case type-checks, and we see that the problem with the [Cons] case is that the pattern-bound variable [idx'] does not have an apparent type compatible with [ls']. In fact, the error message Coq gives for this exact code can be confusing, thanks to an overenthusiastic type inference heuristic. We are told that the [Nil] case body has type [match X with | 0 => A | S _ => unit end] for a unification variable [X], while it is expected to have type [A]. We can see that setting [X] to [0] resolves the conflict, but Coq is not yet smart enough to do this unification automatically. Repeating the function's type in a [return] annotation, used with an [in] annotation, leads us to a more informative error message, saying that [idx'] has type [fin n1] while it is expected to have type [fin n0], where [n0] is bound by the [Cons] pattern and [n1] by the [Next] pattern. As the code is written above, nothing forces these two natural numbers to be equal, though we know intuitively that they must be.
adam@284 84
adam@284 85 We need to use [match] annotations to make the relationship explicit. Unfortunately, the usual trick of postponing argument binding will not help us here. We need to match on both [ls] and [idx]; one or the other must be matched first. To get around this, we apply the convoy pattern that we met last chapter. This application is a little more clever than those we saw before; we use the natural number predecessor function [pred] to express the relationship between the types of these variables.
adamc@106 86
adamc@106 87 [[
adamc@215 88 Fixpoint get n (ls : ilist n) : fin n -> A :=
adamc@215 89 match ls with
adamc@106 90 | Nil => fun idx =>
adamc@215 91 match idx in fin n' return (match n' with
adamc@106 92 | O => A
adamc@106 93 | S _ => unit
adamc@106 94 end) with
adamc@106 95 | First _ => tt
adamc@106 96 | Next _ _ => tt
adamc@106 97 end
adamc@106 98 | Cons _ x ls' => fun idx =>
adamc@215 99 match idx in fin n' return ilist (pred n') -> A with
adamc@106 100 | First _ => fun _ => x
adamc@106 101 | Next _ idx' => fun ls' => get ls' idx'
adamc@106 102 end ls'
adamc@106 103 end.
adamc@106 104
adamc@205 105 ]]
adamc@205 106
adamc@106 107 There is just one problem left with this implementation. Though we know that the local [ls'] in the [Next] case is equal to the original [ls'], the type-checker is not satisfied that the recursive call to [get] does not introduce non-termination. We solve the problem by convoy-binding the partial application of [get] to [ls'], rather than [ls'] by itself. *)
adamc@106 108
adamc@215 109 Fixpoint get n (ls : ilist n) : fin n -> A :=
adamc@215 110 match ls with
adamc@105 111 | Nil => fun idx =>
adamc@215 112 match idx in fin n' return (match n' with
adamc@105 113 | O => A
adamc@105 114 | S _ => unit
adamc@105 115 end) with
adamc@105 116 | First _ => tt
adamc@105 117 | Next _ _ => tt
adamc@105 118 end
adamc@105 119 | Cons _ x ls' => fun idx =>
adamc@215 120 match idx in fin n' return (fin (pred n') -> A) -> A with
adamc@105 121 | First _ => fun _ => x
adamc@105 122 | Next _ idx' => fun get_ls' => get_ls' idx'
adamc@105 123 end (get ls')
adamc@105 124 end.
adamc@113 125 (* end thide *)
adamc@105 126 End ilist.
adamc@105 127
adamc@105 128 Implicit Arguments Nil [A].
adamc@108 129 Implicit Arguments First [n].
adamc@105 130
adamc@108 131 (** A few examples show how to make use of these definitions. *)
adamc@108 132
adamc@108 133 Check Cons 0 (Cons 1 (Cons 2 Nil)).
adamc@215 134 (** %\vspace{-.15in}% [[
adamc@215 135 Cons 0 (Cons 1 (Cons 2 Nil))
adamc@108 136 : ilist nat 3
adam@302 137 ]]
adam@302 138 *)
adamc@215 139
adamc@113 140 (* begin thide *)
adamc@108 141 Eval simpl in get (Cons 0 (Cons 1 (Cons 2 Nil))) First.
adamc@215 142 (** %\vspace{-.15in}% [[
adamc@108 143 = 0
adamc@108 144 : nat
adam@302 145 ]]
adam@302 146 *)
adamc@215 147
adamc@108 148 Eval simpl in get (Cons 0 (Cons 1 (Cons 2 Nil))) (Next First).
adamc@215 149 (** %\vspace{-.15in}% [[
adamc@108 150 = 1
adamc@108 151 : nat
adam@302 152 ]]
adam@302 153 *)
adamc@215 154
adamc@108 155 Eval simpl in get (Cons 0 (Cons 1 (Cons 2 Nil))) (Next (Next First)).
adamc@215 156 (** %\vspace{-.15in}% [[
adamc@108 157 = 2
adamc@108 158 : nat
adam@302 159 ]]
adam@302 160 *)
adamc@113 161 (* end thide *)
adamc@108 162
adamc@108 163 (** Our [get] function is also quite easy to reason about. We show how with a short example about an analogue to the list [map] function. *)
adamc@107 164
adamc@105 165 Section ilist_map.
adamc@105 166 Variables A B : Set.
adamc@105 167 Variable f : A -> B.
adamc@105 168
adamc@215 169 Fixpoint imap n (ls : ilist A n) : ilist B n :=
adamc@215 170 match ls with
adamc@105 171 | Nil => Nil
adamc@105 172 | Cons _ x ls' => Cons (f x) (imap ls')
adamc@105 173 end.
adamc@105 174
adam@406 175 (** It is easy to prove that [get] %``%#"#distributes over#"#%''% [imap] calls. *)
adamc@107 176
adam@342 177 (* EX: Prove that [get] distributes over [imap]. *)
adam@342 178
adam@342 179 (* begin thide *)
adamc@215 180 Theorem get_imap : forall n (idx : fin n) (ls : ilist A n),
adamc@105 181 get (imap ls) idx = f (get ls idx).
adamc@107 182 induction ls; dep_destruct idx; crush.
adamc@105 183 Qed.
adamc@113 184 (* end thide *)
adamc@105 185 End ilist_map.
adamc@107 186
adam@406 187 (** The only tricky bit is remembering to use our [dep_destruct] tactic in place of plain [destruct] when faced with a baffling tactic error message. *)
adamc@107 188
adamc@107 189 (** * Heterogeneous Lists *)
adamc@107 190
adam@342 191 (** Programmers who move to statically typed functional languages from scripting languages often complain about the requirement that every element of a list have the same type. With fancy type systems, we can partially lift this requirement. We can index a list type with a %``%#"#type-level#"#%''% list that explains what type each element of the list should have. This has been done in a variety of ways in Haskell using type classes, and we can do it much more cleanly and directly in Coq. *)
adamc@107 192
adamc@107 193 Section hlist.
adamc@107 194 Variable A : Type.
adamc@107 195 Variable B : A -> Type.
adamc@107 196
adamc@113 197 (* EX: Define a type [hlist] indexed by a [list A], where the type of each element is determined by running [B] on the corresponding element of the index list. *)
adamc@113 198
adam@342 199 (** We parameterize our heterogeneous lists by a type [A] and an [A]-indexed type [B].%\index{Gallina terms!hlist}% *)
adamc@107 200
adamc@113 201 (* begin thide *)
adamc@107 202 Inductive hlist : list A -> Type :=
adamc@107 203 | MNil : hlist nil
adamc@107 204 | MCons : forall (x : A) (ls : list A), B x -> hlist ls -> hlist (x :: ls).
adamc@107 205
adam@342 206 (** 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}% *)
adamc@107 207
adamc@113 208 (* end thide *)
adamc@113 209 (* EX: Define an analogue to [get] for [hlist]s. *)
adamc@113 210
adamc@113 211 (* begin thide *)
adamc@107 212 Variable elm : A.
adamc@107 213
adamc@107 214 Inductive member : list A -> Type :=
adamc@107 215 | MFirst : forall ls, member (elm :: ls)
adamc@107 216 | MNext : forall x ls, member ls -> member (x :: ls).
adamc@107 217
adam@284 218 (** Because the element [elm] that we are %``%#"#searching for#"#%''% in a list does not change across the constructors of [member], we simplify our definitions by making [elm] a local variable. In the definition of [member], we say that [elm] is found in any list that begins with [elm], and, if removing the first element of a list leaves [elm] present, then [elm] is present in the original list, too. The form looks much like a predicate for list membership, but we purposely define [member] in [Type] so that we may decompose its values to guide computations.
adamc@107 219
adam@342 220 We can use [member] to adapt our definition of [get] to [hlist]s. The same basic [match] tricks apply. In the [MCons] case, we form a two-element convoy, passing both the data element [x] and the recursor for the sublist [mls'] to the result of the inner [match]. We did not need to do that in [get]'s definition because the types of list elements were not dependent there. *)
adamc@107 221
adamc@215 222 Fixpoint hget ls (mls : hlist ls) : member ls -> B elm :=
adamc@215 223 match mls with
adamc@107 224 | MNil => fun mem =>
adamc@107 225 match mem in member ls' return (match ls' with
adamc@107 226 | nil => B elm
adamc@107 227 | _ :: _ => unit
adamc@107 228 end) with
adamc@107 229 | MFirst _ => tt
adamc@107 230 | MNext _ _ _ => tt
adamc@107 231 end
adamc@107 232 | MCons _ _ x mls' => fun mem =>
adamc@107 233 match mem in member ls' return (match ls' with
adamc@107 234 | nil => Empty_set
adamc@107 235 | x' :: ls'' =>
adamc@107 236 B x' -> (member ls'' -> B elm) -> B elm
adamc@107 237 end) with
adamc@107 238 | MFirst _ => fun x _ => x
adamc@107 239 | MNext _ _ mem' => fun _ get_mls' => get_mls' mem'
adamc@107 240 end x (hget mls')
adamc@107 241 end.
adamc@113 242 (* end thide *)
adamc@107 243 End hlist.
adamc@108 244
adamc@113 245 (* begin thide *)
adamc@108 246 Implicit Arguments MNil [A B].
adamc@108 247 Implicit Arguments MCons [A B x ls].
adamc@108 248
adamc@108 249 Implicit Arguments MFirst [A elm ls].
adamc@108 250 Implicit Arguments MNext [A elm x ls].
adamc@113 251 (* end thide *)
adamc@108 252
adamc@108 253 (** 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. *)
adamc@108 254
adamc@108 255 Definition someTypes : list Set := nat :: bool :: nil.
adamc@108 256
adamc@113 257 (* begin thide *)
adamc@113 258
adamc@108 259 Example someValues : hlist (fun T : Set => T) someTypes :=
adamc@108 260 MCons 5 (MCons true MNil).
adamc@108 261
adamc@108 262 Eval simpl in hget someValues MFirst.
adamc@215 263 (** %\vspace{-.15in}% [[
adamc@108 264 = 5
adamc@108 265 : (fun T : Set => T) nat
adam@302 266 ]]
adam@302 267 *)
adamc@215 268
adamc@108 269 Eval simpl in hget someValues (MNext MFirst).
adamc@215 270 (** %\vspace{-.15in}% [[
adamc@108 271 = true
adamc@108 272 : (fun T : Set => T) bool
adam@302 273 ]]
adam@302 274 *)
adamc@108 275
adamc@108 276 (** We can also build indexed lists of pairs in this way. *)
adamc@108 277
adamc@108 278 Example somePairs : hlist (fun T : Set => T * T)%type someTypes :=
adamc@108 279 MCons (1, 2) (MCons (true, false) MNil).
adamc@108 280
adamc@113 281 (* end thide *)
adamc@113 282
adamc@113 283
adamc@108 284 (** ** A Lambda Calculus Interpreter *)
adamc@108 285
adam@342 286 (** Heterogeneous lists are very useful in implementing %\index{interpreters}%interpreters for functional programming languages. Using the types and operations we have already defined, it is trivial to write an interpreter for simply typed lambda calculus%\index{lambda calculus}%. Our interpreter can alternatively be thought of as a denotational semantics.
adamc@108 287
adamc@108 288 We start with an algebraic datatype for types. *)
adamc@108 289
adamc@108 290 Inductive type : Set :=
adamc@108 291 | Unit : type
adamc@108 292 | Arrow : type -> type -> type.
adamc@108 293
adam@342 294 (** Now we can define a type family for expressions. An [exp ts t] will stand for an expression that has type [t] and whose free variables have types in the list [ts]. We effectively use the de Bruijn index variable representation%~\cite{DeBruijn}%. Variables are represented as [member] values; that is, a variable is more or less a constructive proof that a particular type is found in the type environment. *)
adamc@108 295
adamc@108 296 Inductive exp : list type -> type -> Set :=
adamc@108 297 | Const : forall ts, exp ts Unit
adamc@113 298 (* begin thide *)
adamc@108 299 | Var : forall ts t, member t ts -> exp ts t
adamc@108 300 | App : forall ts dom ran, exp ts (Arrow dom ran) -> exp ts dom -> exp ts ran
adamc@108 301 | Abs : forall ts dom ran, exp (dom :: ts) ran -> exp ts (Arrow dom ran).
adamc@113 302 (* end thide *)
adamc@108 303
adamc@108 304 Implicit Arguments Const [ts].
adamc@108 305
adamc@108 306 (** We write a simple recursive function to translate [type]s into [Set]s. *)
adamc@108 307
adamc@108 308 Fixpoint typeDenote (t : type) : Set :=
adamc@108 309 match t with
adamc@108 310 | Unit => unit
adamc@108 311 | Arrow t1 t2 => typeDenote t1 -> typeDenote t2
adamc@108 312 end.
adamc@108 313
adam@342 314 (** Now it is straightforward to write an expression interpreter. The type of the function, [expDenote], tells us that we translate expressions into functions from properly typed environments to final values. An environment for a free variable list [ts] is simply a [hlist typeDenote ts]. That is, for each free variable, the heterogeneous list that is the environment must have a value of the variable's associated type. We use [hget] to implement the [Var] case, and we use [MCons] to extend the environment in the [Abs] case. *)
adamc@108 315
adamc@113 316 (* EX: Define an interpreter for [exp]s. *)
adamc@113 317
adamc@113 318 (* begin thide *)
adamc@215 319 Fixpoint expDenote ts t (e : exp ts t) : hlist typeDenote ts -> typeDenote t :=
adamc@215 320 match e with
adamc@108 321 | Const _ => fun _ => tt
adamc@108 322
adamc@108 323 | Var _ _ mem => fun s => hget s mem
adamc@108 324 | App _ _ _ e1 e2 => fun s => (expDenote e1 s) (expDenote e2 s)
adamc@108 325 | Abs _ _ _ e' => fun s => fun x => expDenote e' (MCons x s)
adamc@108 326 end.
adamc@108 327
adamc@108 328 (** Like for previous examples, our interpreter is easy to run with [simpl]. *)
adamc@108 329
adamc@108 330 Eval simpl in expDenote Const MNil.
adamc@215 331 (** %\vspace{-.15in}% [[
adamc@108 332 = tt
adamc@108 333 : typeDenote Unit
adam@302 334 ]]
adam@302 335 *)
adamc@215 336
adamc@108 337 Eval simpl in expDenote (Abs (dom := Unit) (Var MFirst)) MNil.
adamc@215 338 (** %\vspace{-.15in}% [[
adamc@108 339 = fun x : unit => x
adamc@108 340 : typeDenote (Arrow Unit Unit)
adam@302 341 ]]
adam@302 342 *)
adamc@215 343
adamc@108 344 Eval simpl in expDenote (Abs (dom := Unit)
adamc@108 345 (Abs (dom := Unit) (Var (MNext MFirst)))) MNil.
adamc@215 346 (** %\vspace{-.15in}% [[
adamc@108 347 = fun x _ : unit => x
adamc@108 348 : typeDenote (Arrow Unit (Arrow Unit Unit))
adam@302 349 ]]
adam@302 350 *)
adamc@215 351
adamc@108 352 Eval simpl in expDenote (Abs (dom := Unit) (Abs (dom := Unit) (Var MFirst))) MNil.
adamc@215 353 (** %\vspace{-.15in}% [[
adamc@108 354 = fun _ x0 : unit => x0
adamc@108 355 : typeDenote (Arrow Unit (Arrow Unit Unit))
adam@302 356 ]]
adam@302 357 *)
adamc@215 358
adamc@108 359 Eval simpl in expDenote (App (Abs (Var MFirst)) Const) MNil.
adamc@215 360 (** %\vspace{-.15in}% [[
adamc@108 361 = tt
adamc@108 362 : typeDenote Unit
adam@302 363 ]]
adam@302 364 *)
adamc@108 365
adamc@113 366 (* end thide *)
adamc@113 367
adam@342 368 (** We are starting to develop the tools behind dependent typing's amazing advantage over alternative approaches in several important areas. Here, we have implemented complete syntax, typing rules, and evaluation semantics for simply typed lambda calculus without even needing to define a syntactic substitution operation. We did it all without a single line of proof, and our implementation is manifestly executable. Other, more common approaches to language formalization often state and prove explicit theorems about type safety of languages. In the above example, we got type safety, termination, and other meta-theorems for free, by reduction to CIC, which we know has those properties. *)
adamc@108 369
adamc@108 370
adamc@109 371 (** * Recursive Type Definitions *)
adamc@109 372
adam@398 373 (** %\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. *)
adamc@109 374
adamc@113 375 (* EX: Come up with an alternate [ilist] definition that makes it easier to write [get]. *)
adamc@113 376
adamc@109 377 Section filist.
adamc@109 378 Variable A : Set.
adamc@109 379
adamc@113 380 (* begin thide *)
adamc@109 381 Fixpoint filist (n : nat) : Set :=
adamc@109 382 match n with
adamc@109 383 | O => unit
adamc@109 384 | S n' => A * filist n'
adamc@109 385 end%type.
adamc@109 386
adamc@109 387 (** We say that a list of length 0 has no contents, and a list of length [S n'] is a pair of a data value and a list of length [n']. *)
adamc@109 388
adamc@215 389 Fixpoint ffin (n : nat) : Set :=
adamc@109 390 match n with
adamc@109 391 | O => Empty_set
adamc@215 392 | S n' => option (ffin n')
adamc@109 393 end.
adamc@109 394
adam@406 395 (** We express that there are no index values when [n = O], by defining such indices as type [Empty_set]; and we express that, at [n = S n'], there is a choice between picking the first element of the list (represented as [None]) or choosing a later element (represented by [Some idx], where [idx] is an index into the list tail). For instance, the three values of type [ffin 3] are [None], [Some None], and [Some (Some None)]. *)
adamc@109 396
adamc@215 397 Fixpoint fget (n : nat) : filist n -> ffin n -> A :=
adamc@215 398 match n with
adamc@109 399 | O => fun _ idx => match idx with end
adamc@109 400 | S n' => fun ls idx =>
adamc@109 401 match idx with
adamc@109 402 | None => fst ls
adamc@109 403 | Some idx' => fget n' (snd ls) idx'
adamc@109 404 end
adamc@109 405 end.
adamc@109 406
adamc@215 407 (** Our new [get] implementation needs only one dependent [match], and its annotation is inferred for us. Our choices of data structure implementations lead to just the right typing behavior for this new definition to work out. *)
adamc@113 408 (* end thide *)
adamc@215 409
adamc@109 410 End filist.
adamc@109 411
adamc@109 412 (** Heterogeneous lists are a little trickier to define with recursion, but we then reap similar benefits in simplicity of use. *)
adamc@109 413
adamc@113 414 (* EX: Come up with an alternate [hlist] definition that makes it easier to write [hget]. *)
adamc@113 415
adamc@109 416 Section fhlist.
adamc@109 417 Variable A : Type.
adamc@109 418 Variable B : A -> Type.
adamc@109 419
adamc@113 420 (* begin thide *)
adamc@109 421 Fixpoint fhlist (ls : list A) : Type :=
adamc@109 422 match ls with
adamc@109 423 | nil => unit
adamc@109 424 | x :: ls' => B x * fhlist ls'
adamc@109 425 end%type.
adamc@109 426
adam@342 427 (** The definition of [fhlist] follows the definition of [filist], with the added wrinkle of dependently typed data elements. *)
adamc@109 428
adamc@109 429 Variable elm : A.
adamc@109 430
adamc@109 431 Fixpoint fmember (ls : list A) : Type :=
adamc@109 432 match ls with
adamc@109 433 | nil => Empty_set
adamc@109 434 | x :: ls' => (x = elm) + fmember ls'
adamc@109 435 end%type.
adamc@109 436
adamc@215 437 (** The definition of [fmember] follows the definition of [ffin]. Empty lists have no members, and member types for nonempty lists are built by adding one new option to the type of members of the list tail. While for [index] we needed no new information associated with the option that we add, here we need to know that the head of the list equals the element we are searching for. We express that with a sum type whose left branch is the appropriate equality proposition. Since we define [fmember] to live in [Type], we can insert [Prop] types as needed, because [Prop] is a subtype of [Type].
adamc@109 438
adamc@109 439 We know all of the tricks needed to write a first attempt at a [get] function for [fhlist]s.
adamc@109 440
adamc@109 441 [[
adamc@109 442 Fixpoint fhget (ls : list A) : fhlist ls -> fmember ls -> B elm :=
adamc@215 443 match ls with
adamc@109 444 | nil => fun _ idx => match idx with end
adamc@109 445 | _ :: ls' => fun mls idx =>
adamc@109 446 match idx with
adamc@109 447 | inl _ => fst mls
adamc@109 448 | inr idx' => fhget ls' (snd mls) idx'
adamc@109 449 end
adamc@109 450 end.
adamc@109 451
adamc@205 452 ]]
adamc@205 453
adamc@109 454 Only one problem remains. The expression [fst mls] is not known to have the proper type. To demonstrate that it does, we need to use the proof available in the [inl] case of the inner [match]. *)
adamc@109 455
adamc@109 456 Fixpoint fhget (ls : list A) : fhlist ls -> fmember ls -> B elm :=
adamc@215 457 match ls with
adamc@109 458 | nil => fun _ idx => match idx with end
adamc@109 459 | _ :: ls' => fun mls idx =>
adamc@109 460 match idx with
adamc@109 461 | inl pf => match pf with
adamc@109 462 | refl_equal => fst mls
adamc@109 463 end
adamc@109 464 | inr idx' => fhget ls' (snd mls) idx'
adamc@109 465 end
adamc@109 466 end.
adamc@109 467
adamc@109 468 (** By pattern-matching on the equality proof [pf], we make that equality known to the type-checker. Exactly why this works can be seen by studying the definition of equality. *)
adamc@109 469
adamc@109 470 Print eq.
adamc@215 471 (** %\vspace{-.15in}% [[
adamc@109 472 Inductive eq (A : Type) (x : A) : A -> Prop := refl_equal : x = x
adamc@215 473
adamc@109 474 ]]
adamc@109 475
adamc@215 476 In a proposition [x = y], we see that [x] is a parameter and [y] is a regular argument. The type of the constructor [refl_equal] shows that [y] can only ever be instantiated to [x]. Thus, within a pattern-match with [refl_equal], occurrences of [y] can be replaced with occurrences of [x] for typing purposes. *)
adamc@113 477 (* end thide *)
adamc@215 478
adamc@109 479 End fhlist.
adamc@110 480
adamc@111 481 Implicit Arguments fhget [A B elm ls].
adamc@111 482
adamc@110 483
adamc@110 484 (** * Data Structures as Index Functions *)
adamc@110 485
adam@342 486 (** %\index{index function}%Indexed lists can be useful in defining other inductive types with constructors that take variable numbers of arguments. In this section, we consider parameterized trees with arbitrary branching factor. *)
adamc@110 487
adamc@110 488 Section tree.
adamc@110 489 Variable A : Set.
adamc@110 490
adamc@110 491 Inductive tree : Set :=
adamc@110 492 | Leaf : A -> tree
adamc@110 493 | Node : forall n, ilist tree n -> tree.
adamc@110 494 End tree.
adamc@110 495
adamc@110 496 (** Every [Node] of a [tree] has a natural number argument, which gives the number of child trees in the second argument, typed with [ilist]. We can define two operations on trees of naturals: summing their elements and incrementing their elements. It is useful to define a generic fold function on [ilist]s first. *)
adamc@110 497
adamc@110 498 Section ifoldr.
adamc@110 499 Variables A B : Set.
adamc@110 500 Variable f : A -> B -> B.
adamc@110 501 Variable i : B.
adamc@110 502
adamc@215 503 Fixpoint ifoldr n (ls : ilist A n) : B :=
adamc@110 504 match ls with
adamc@110 505 | Nil => i
adamc@110 506 | Cons _ x ls' => f x (ifoldr ls')
adamc@110 507 end.
adamc@110 508 End ifoldr.
adamc@110 509
adamc@110 510 Fixpoint sum (t : tree nat) : nat :=
adamc@110 511 match t with
adamc@110 512 | Leaf n => n
adamc@110 513 | Node _ ls => ifoldr (fun t' n => sum t' + n) O ls
adamc@110 514 end.
adamc@110 515
adamc@110 516 Fixpoint inc (t : tree nat) : tree nat :=
adamc@110 517 match t with
adamc@110 518 | Leaf n => Leaf (S n)
adamc@110 519 | Node _ ls => Node (imap inc ls)
adamc@110 520 end.
adamc@110 521
adamc@110 522 (** Now we might like to prove that [inc] does not decrease a tree's [sum]. *)
adamc@110 523
adamc@110 524 Theorem sum_inc : forall t, sum (inc t) >= sum t.
adamc@113 525 (* begin thide *)
adamc@110 526 induction t; crush.
adamc@110 527 (** [[
adamc@110 528 n : nat
adamc@110 529 i : ilist (tree nat) n
adamc@110 530 ============================
adamc@110 531 ifoldr (fun (t' : tree nat) (n0 : nat) => sum t' + n0) 0 (imap inc i) >=
adamc@110 532 ifoldr (fun (t' : tree nat) (n0 : nat) => sum t' + n0) 0 i
adamc@215 533
adamc@110 534 ]]
adamc@110 535
adam@342 536 We are left with a single subgoal which does not seem provable directly. This is the same problem that we met in Chapter 3 with other %\index{nested inductive type}%nested inductive types. *)
adamc@110 537
adamc@110 538 Check tree_ind.
adamc@215 539 (** %\vspace{-.15in}% [[
adamc@215 540 tree_ind
adamc@110 541 : forall (A : Set) (P : tree A -> Prop),
adamc@110 542 (forall a : A, P (Leaf a)) ->
adamc@110 543 (forall (n : nat) (i : ilist (tree A) n), P (Node i)) ->
adamc@110 544 forall t : tree A, P t
adamc@215 545
adamc@110 546 ]]
adamc@110 547
adam@342 548 The automatically generated induction principle is too weak. For the [Node] case, it gives us no inductive hypothesis. We could write our own induction principle, as we did in Chapter 3, but there is an easier way, if we are willing to alter the definition of [tree]. *)
adamc@215 549
adamc@110 550 Abort.
adamc@110 551
adamc@110 552 Reset tree.
adamc@110 553
adamc@110 554 (** First, let us try using our recursive definition of [ilist]s instead of the inductive version. *)
adamc@110 555
adamc@110 556 Section tree.
adamc@110 557 Variable A : Set.
adamc@110 558
adamc@215 559 (** %\vspace{-.15in}% [[
adamc@110 560 Inductive tree : Set :=
adamc@110 561 | Leaf : A -> tree
adamc@110 562 | Node : forall n, filist tree n -> tree.
adam@342 563 ]]
adamc@110 564
adam@342 565 <<
adamc@110 566 Error: Non strictly positive occurrence of "tree" in
adamc@110 567 "forall n : nat, filist tree n -> tree"
adam@342 568 >>
adamc@110 569
adam@342 570 The special-case rule for nested datatypes only works with nested uses of other inductive types, which could be replaced with uses of new mutually inductive types. We defined [filist] recursively, so it may not be used for nested recursion.
adamc@110 571
adam@398 572 Our final solution uses yet another of the inductive definition techniques introduced in Chapter 3, %\index{reflexive inductive type}%reflexive types. Instead of merely using [fin] to get elements out of [ilist], we can _define_ [ilist] in terms of [fin]. For the reasons outlined above, it turns out to be easier to work with [ffin] in place of [fin]. *)
adamc@110 573
adamc@110 574 Inductive tree : Set :=
adamc@110 575 | Leaf : A -> tree
adamc@215 576 | Node : forall n, (ffin n -> tree) -> tree.
adamc@110 577
adamc@215 578 (** A [Node] is indexed by a natural number [n], and the node's [n] children are represented as a function from [ffin n] to trees, which is isomorphic to the [ilist]-based representation that we used above. *)
adamc@215 579
adamc@110 580 End tree.
adamc@110 581
adamc@110 582 Implicit Arguments Node [A n].
adamc@110 583
adamc@215 584 (** We can redefine [sum] and [inc] for our new [tree] type. Again, it is useful to define a generic fold function first. This time, it takes in a function whose range is some [ffin] type, and it folds another function over the results of calling the first function at every possible [ffin] value. *)
adamc@110 585
adamc@110 586 Section rifoldr.
adamc@110 587 Variables A B : Set.
adamc@110 588 Variable f : A -> B -> B.
adamc@110 589 Variable i : B.
adamc@110 590
adamc@215 591 Fixpoint rifoldr (n : nat) : (ffin n -> A) -> B :=
adamc@215 592 match n with
adamc@110 593 | O => fun _ => i
adamc@110 594 | S n' => fun get => f (get None) (rifoldr n' (fun idx => get (Some idx)))
adamc@110 595 end.
adamc@110 596 End rifoldr.
adamc@110 597
adamc@110 598 Implicit Arguments rifoldr [A B n].
adamc@110 599
adamc@110 600 Fixpoint sum (t : tree nat) : nat :=
adamc@110 601 match t with
adamc@110 602 | Leaf n => n
adamc@110 603 | Node _ f => rifoldr plus O (fun idx => sum (f idx))
adamc@110 604 end.
adamc@110 605
adamc@110 606 Fixpoint inc (t : tree nat) : tree nat :=
adamc@110 607 match t with
adamc@110 608 | Leaf n => Leaf (S n)
adamc@110 609 | Node _ f => Node (fun idx => inc (f idx))
adamc@110 610 end.
adamc@110 611
adam@398 612 (** Now we are ready to prove the theorem where we got stuck before. We will not need to define any new induction principle, but it _will_ be helpful to prove some lemmas. *)
adamc@110 613
adamc@110 614 Lemma plus_ge : forall x1 y1 x2 y2,
adamc@110 615 x1 >= x2
adamc@110 616 -> y1 >= y2
adamc@110 617 -> x1 + y1 >= x2 + y2.
adamc@110 618 crush.
adamc@110 619 Qed.
adamc@110 620
adamc@215 621 Lemma sum_inc' : forall n (f1 f2 : ffin n -> nat),
adamc@110 622 (forall idx, f1 idx >= f2 idx)
adamc@110 623 -> rifoldr plus 0 f1 >= rifoldr plus 0 f2.
adamc@110 624 Hint Resolve plus_ge.
adamc@110 625
adamc@110 626 induction n; crush.
adamc@110 627 Qed.
adamc@110 628
adamc@110 629 Theorem sum_inc : forall t, sum (inc t) >= sum t.
adamc@110 630 Hint Resolve sum_inc'.
adamc@110 631
adamc@110 632 induction t; crush.
adamc@110 633 Qed.
adamc@110 634
adamc@113 635 (* end thide *)
adamc@113 636
adamc@110 637 (** Even if Coq would generate complete induction principles automatically for nested inductive definitions like the one we started with, there would still be advantages to using this style of reflexive encoding. We see one of those advantages in the definition of [inc], where we did not need to use any kind of auxiliary function. In general, reflexive encodings often admit direct implementations of operations that would require recursion if performed with more traditional inductive data structures. *)
adamc@111 638
adamc@111 639 (** ** Another Interpreter Example *)
adamc@111 640
adam@342 641 (** We develop another example of variable-arity constructors, in the form of optimization of a small expression language with a construct like Scheme's %\texttt{%#<tt>#cond#</tt>#%}%. Each of our conditional expressions takes a list of pairs of boolean tests and bodies. The value of the conditional comes from the body of the first test in the list to evaluate to [true]. To simplify the %\index{interpreters}%interpreter we will write, we force each conditional to include a final, default case. *)
adamc@112 642
adamc@112 643 Inductive type' : Type := Nat | Bool.
adamc@111 644
adamc@111 645 Inductive exp' : type' -> Type :=
adamc@112 646 | NConst : nat -> exp' Nat
adamc@112 647 | Plus : exp' Nat -> exp' Nat -> exp' Nat
adamc@112 648 | Eq : exp' Nat -> exp' Nat -> exp' Bool
adamc@111 649
adamc@112 650 | BConst : bool -> exp' Bool
adamc@113 651 (* begin thide *)
adamc@215 652 | Cond : forall n t, (ffin n -> exp' Bool)
adamc@215 653 -> (ffin n -> exp' t) -> exp' t -> exp' t.
adamc@113 654 (* end thide *)
adamc@111 655
adam@284 656 (** A [Cond] is parameterized by a natural [n], which tells us how many cases this conditional has. The test expressions are represented with a function of type [ffin n -> exp' Bool], and the bodies are represented with a function of type [ffin n -> exp' t], where [t] is the overall type. The final [exp' t] argument is the default case. For example, here is an expression that successively checks whether [2 + 2 = 5] (returning 0 if so) or if [1 + 1 = 2] (returning 1 if so), returning 2 otherwise. *)
adamc@112 657
adam@284 658 Example ex1 := Cond 2
adam@284 659 (fun f => match f with
adam@284 660 | None => Eq (Plus (NConst 2) (NConst 2)) (NConst 5)
adam@284 661 | Some None => Eq (Plus (NConst 1) (NConst 1)) (NConst 2)
adam@284 662 | Some (Some v) => match v with end
adam@284 663 end)
adam@284 664 (fun f => match f with
adam@284 665 | None => NConst 0
adam@284 666 | Some None => NConst 1
adam@284 667 | Some (Some v) => match v with end
adam@284 668 end)
adam@284 669 (NConst 2).
adam@284 670
adam@284 671 (** We start implementing our interpreter with a standard type denotation function. *)
adamc@112 672
adamc@111 673 Definition type'Denote (t : type') : Set :=
adamc@111 674 match t with
adamc@112 675 | Nat => nat
adamc@112 676 | Bool => bool
adamc@111 677 end.
adamc@111 678
adamc@112 679 (** To implement the expression interpreter, it is useful to have the following function that implements the functionality of [Cond] without involving any syntax. *)
adamc@112 680
adamc@113 681 (* begin thide *)
adamc@111 682 Section cond.
adamc@111 683 Variable A : Set.
adamc@111 684 Variable default : A.
adamc@111 685
adamc@215 686 Fixpoint cond (n : nat) : (ffin n -> bool) -> (ffin n -> A) -> A :=
adamc@215 687 match n with
adamc@111 688 | O => fun _ _ => default
adamc@111 689 | S n' => fun tests bodies =>
adamc@111 690 if tests None
adamc@111 691 then bodies None
adamc@111 692 else cond n'
adamc@111 693 (fun idx => tests (Some idx))
adamc@111 694 (fun idx => bodies (Some idx))
adamc@111 695 end.
adamc@111 696 End cond.
adamc@111 697
adamc@111 698 Implicit Arguments cond [A n].
adamc@113 699 (* end thide *)
adamc@111 700
adamc@112 701 (** Now the expression interpreter is straightforward to write. *)
adamc@112 702
adamc@215 703 Fixpoint exp'Denote t (e : exp' t) : type'Denote t :=
adamc@215 704 match e with
adamc@215 705 | NConst n => n
adamc@215 706 | Plus e1 e2 => exp'Denote e1 + exp'Denote e2
adamc@111 707 | Eq e1 e2 =>
adamc@111 708 if eq_nat_dec (exp'Denote e1) (exp'Denote e2) then true else false
adamc@111 709
adamc@215 710 | BConst b => b
adamc@111 711 | Cond _ _ tests bodies default =>
adamc@113 712 (* begin thide *)
adamc@111 713 cond
adamc@111 714 (exp'Denote default)
adamc@111 715 (fun idx => exp'Denote (tests idx))
adamc@111 716 (fun idx => exp'Denote (bodies idx))
adamc@113 717 (* end thide *)
adamc@111 718 end.
adamc@111 719
adamc@112 720 (** We will implement a constant-folding function that optimizes conditionals, removing cases with known-[false] tests and cases that come after known-[true] tests. A function [cfoldCond] implements the heart of this logic. The convoy pattern is used again near the end of the implementation. *)
adamc@112 721
adamc@113 722 (* begin thide *)
adamc@111 723 Section cfoldCond.
adamc@111 724 Variable t : type'.
adamc@111 725 Variable default : exp' t.
adamc@111 726
adamc@112 727 Fixpoint cfoldCond (n : nat)
adamc@215 728 : (ffin n -> exp' Bool) -> (ffin n -> exp' t) -> exp' t :=
adamc@215 729 match n with
adamc@111 730 | O => fun _ _ => default
adamc@111 731 | S n' => fun tests bodies =>
adamc@204 732 match tests None return _ with
adamc@111 733 | BConst true => bodies None
adamc@111 734 | BConst false => cfoldCond n'
adamc@111 735 (fun idx => tests (Some idx))
adamc@111 736 (fun idx => bodies (Some idx))
adamc@111 737 | _ =>
adamc@111 738 let e := cfoldCond n'
adamc@111 739 (fun idx => tests (Some idx))
adamc@111 740 (fun idx => bodies (Some idx)) in
adamc@112 741 match e in exp' t return exp' t -> exp' t with
adamc@112 742 | Cond n _ tests' bodies' default' => fun body =>
adamc@111 743 Cond
adamc@111 744 (S n)
adamc@111 745 (fun idx => match idx with
adamc@112 746 | None => tests None
adamc@111 747 | Some idx => tests' idx
adamc@111 748 end)
adamc@111 749 (fun idx => match idx with
adamc@111 750 | None => body
adamc@111 751 | Some idx => bodies' idx
adamc@111 752 end)
adamc@111 753 default'
adamc@112 754 | e => fun body =>
adamc@111 755 Cond
adamc@111 756 1
adamc@112 757 (fun _ => tests None)
adamc@111 758 (fun _ => body)
adamc@111 759 e
adamc@112 760 end (bodies None)
adamc@111 761 end
adamc@111 762 end.
adamc@111 763 End cfoldCond.
adamc@111 764
adamc@111 765 Implicit Arguments cfoldCond [t n].
adamc@113 766 (* end thide *)
adamc@111 767
adamc@112 768 (** Like for the interpreters, most of the action was in this helper function, and [cfold] itself is easy to write. *)
adamc@112 769
adamc@215 770 Fixpoint cfold t (e : exp' t) : exp' t :=
adamc@215 771 match e with
adamc@111 772 | NConst n => NConst n
adamc@111 773 | Plus e1 e2 =>
adamc@111 774 let e1' := cfold e1 in
adamc@111 775 let e2' := cfold e2 in
adam@417 776 match e1', e2' return exp' Nat with
adamc@111 777 | NConst n1, NConst n2 => NConst (n1 + n2)
adamc@111 778 | _, _ => Plus e1' e2'
adamc@111 779 end
adamc@111 780 | Eq e1 e2 =>
adamc@111 781 let e1' := cfold e1 in
adamc@111 782 let e2' := cfold e2 in
adam@417 783 match e1', e2' return exp' Bool with
adamc@111 784 | NConst n1, NConst n2 => BConst (if eq_nat_dec n1 n2 then true else false)
adamc@111 785 | _, _ => Eq e1' e2'
adamc@111 786 end
adamc@111 787
adamc@111 788 | BConst b => BConst b
adamc@111 789 | Cond _ _ tests bodies default =>
adamc@113 790 (* begin thide *)
adamc@111 791 cfoldCond
adamc@111 792 (cfold default)
adamc@111 793 (fun idx => cfold (tests idx))
adamc@111 794 (fun idx => cfold (bodies idx))
adamc@113 795 (* end thide *)
adamc@111 796 end.
adamc@111 797
adamc@113 798 (* begin thide *)
adam@342 799 (** To prove our final correctness theorem, it is useful to know that [cfoldCond] preserves expression meanings. This lemma formalizes that property. The proof is a standard mostly automated one, with the only wrinkle being a guided instantiation of the quantifiers in the induction hypothesis. *)
adamc@112 800
adamc@111 801 Lemma cfoldCond_correct : forall t (default : exp' t)
adamc@215 802 n (tests : ffin n -> exp' Bool) (bodies : ffin n -> exp' t),
adamc@111 803 exp'Denote (cfoldCond default tests bodies)
adamc@111 804 = exp'Denote (Cond n tests bodies default).
adamc@111 805 induction n; crush;
adamc@111 806 match goal with
adamc@111 807 | [ IHn : forall tests bodies, _, tests : _ -> _, bodies : _ -> _ |- _ ] =>
adam@294 808 specialize (IHn (fun idx => tests (Some idx)) (fun idx => bodies (Some idx)))
adamc@111 809 end;
adamc@111 810 repeat (match goal with
adam@406 811 | [ |- context[match ?E with NConst _ => _ | _ => _ end] ] => dep_destruct E
adamc@111 812 | [ |- context[if ?B then _ else _] ] => destruct B
adamc@111 813 end; crush).
adamc@111 814 Qed.
adamc@111 815
adam@398 816 (** It is also useful to know that the result of a call to [cond] is not changed by substituting new tests and bodies functions, so long as the new functions have the same input-output behavior as the old. It turns out that, in Coq, it is not possible to prove in general that functions related in this way are equal. We treat this issue with our discussion of axioms in a later chapter. For now, it suffices to prove that the particular function [cond] is _extensional_; that is, it is unaffected by substitution of functions with input-output equivalents. *)
adamc@112 817
adamc@215 818 Lemma cond_ext : forall (A : Set) (default : A) n (tests tests' : ffin n -> bool)
adamc@215 819 (bodies bodies' : ffin n -> A),
adamc@111 820 (forall idx, tests idx = tests' idx)
adamc@111 821 -> (forall idx, bodies idx = bodies' idx)
adamc@111 822 -> cond default tests bodies
adamc@111 823 = cond default tests' bodies'.
adamc@111 824 induction n; crush;
adamc@111 825 match goal with
adamc@111 826 | [ |- context[if ?E then _ else _] ] => destruct E
adamc@111 827 end; crush.
adamc@111 828 Qed.
adamc@111 829
adamc@112 830 (** Now the final theorem is easy to prove. We add our two lemmas as hints and perform standard automation with pattern-matching of subterms to destruct. *)
adamc@113 831 (* end thide *)
adamc@112 832
adamc@111 833 Theorem cfold_correct : forall t (e : exp' t),
adamc@111 834 exp'Denote (cfold e) = exp'Denote e.
adamc@113 835 (* begin thide *)
adam@375 836 Hint Rewrite cfoldCond_correct.
adamc@111 837 Hint Resolve cond_ext.
adamc@111 838
adamc@111 839 induction e; crush;
adamc@111 840 repeat (match goal with
adamc@111 841 | [ |- context[cfold ?E] ] => dep_destruct (cfold E)
adamc@111 842 end; crush).
adamc@111 843 Qed.
adamc@113 844 (* end thide *)
adamc@115 845
adamc@115 846
adamc@215 847 (** * Choosing Between Representations *)
adamc@215 848
adamc@215 849 (** It is not always clear which of these representation techniques to apply in a particular situation, but I will try to summarize the pros and cons of each.
adamc@215 850
adamc@215 851 Inductive types are often the most pleasant to work with, after someone has spent the time implementing some basic library functions for them, using fancy [match] annotations. Many aspects of Coq's logic and tactic support are specialized to deal with inductive types, and you may miss out if you use alternate encodings.
adamc@215 852
adam@406 853 Recursive types usually involve much less initial effort, but they can be less convenient to use with proof automation. For instance, the [simpl] tactic (which is among the ingredients in [crush]) will sometimes be overzealous in simplifying uses of functions over recursive types. Consider a call [get l f], where variable [l] has type [filist A (S n)]. The type of [l] would be simplified to an explicit pair type. In a proof involving many recursive types, this kind of unhelpful %``%#"#simplification#"#%''% can lead to rapid bloat in the sizes of subgoals. Even worse, it can prevent syntactic pattern-matching, like in cases where [filist] is expected but a pair type is found in the %``%#"#simplified#"#%''% version. The same problem applies to applications of recursive functions to values in recursive types: the recursive function call may %``%#"#simplify#"#%''% when the top-level structure of the type index but not the recursive value is known, because such functions are generally defined by recursion on the index, not the value.
adamc@215 854
adam@342 855 Another disadvantage of recursive types is that they only apply to type families whose indices determine their %``%#"#skeletons.#"#%''% This is not true for all data structures; a good counterexample comes from the richly typed programming language syntax types we have used several times so far. The fact that a piece of syntax has type [Nat] tells us nothing about the tree structure of that syntax.
adamc@215 856
adam@342 857 Finally, Coq type inference can be more helpful in constructing values in inductive types. Application of a particular constructor of that type tells Coq what to expect from the arguments, while, for instance, forming a generic pair does not make clear an intention to interpret the value as belonging to a particular recursive type. This downside can be mitigated to an extent by writing %``%#"#constructor#"#%''% functions for a recursive type, mirroring the definition of the corresponding inductive type.
adam@342 858
adam@342 859 Reflexive encodings of data types are seen relatively rarely. As our examples demonstrated, manipulating index values manually can lead to hard-to-read code. A normal inductive type is generally easier to work with, once someone has gone through the trouble of implementing an induction principle manually with the techniques we studied in Chapter 3. For small developments, avoiding that kind of coding can justify the use of reflexive data structures. There are also some useful instances of %\index{co-inductive types}%co-inductive definitions with nested data structures (e.g., lists of values in the co-inductive type) that can only be deconstructed effectively with reflexive encoding of the nested structures. *)