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author Adam Chlipala <adam@chlipala.net>
date Wed, 02 Nov 2011 17:03:25 -0400
parents 25d60fed2e96
children 549d604c3d16
rev   line source
adam@342 1 (* Copyright (c) 2008-2011, 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@342 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@284 175 (** It is easy to prove that [get] %``%#"#distributes over#"#%''% [imap] calls. 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@107 176
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
adamc@107 187
adamc@107 188 (** * Heterogeneous Lists *)
adamc@107 189
adam@342 190 (** 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@107 191
adamc@107 192 Section hlist.
adamc@107 193 Variable A : Type.
adamc@107 194 Variable B : A -> Type.
adamc@107 195
adamc@113 196 (* 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@113 197
adam@342 198 (** We parameterize our heterogeneous lists by a type [A] and an [A]-indexed type [B].%\index{Gallina terms!hlist}% *)
adamc@107 199
adamc@113 200 (* begin thide *)
adamc@107 201 Inductive hlist : list A -> Type :=
adamc@107 202 | MNil : hlist nil
adamc@107 203 | MCons : forall (x : A) (ls : list A), B x -> hlist ls -> hlist (x :: ls).
adamc@107 204
adam@342 205 (** 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@107 206
adamc@113 207 (* end thide *)
adamc@113 208 (* EX: Define an analogue to [get] for [hlist]s. *)
adamc@113 209
adamc@113 210 (* begin thide *)
adamc@107 211 Variable elm : A.
adamc@107 212
adamc@107 213 Inductive member : list A -> Type :=
adamc@107 214 | MFirst : forall ls, member (elm :: ls)
adamc@107 215 | MNext : forall x ls, member ls -> member (x :: ls).
adamc@107 216
adam@284 217 (** 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@107 218
adam@342 219 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@107 220
adamc@215 221 Fixpoint hget ls (mls : hlist ls) : member ls -> B elm :=
adamc@215 222 match mls with
adamc@107 223 | MNil => fun mem =>
adamc@107 224 match mem in member ls' return (match ls' with
adamc@107 225 | nil => B elm
adamc@107 226 | _ :: _ => unit
adamc@107 227 end) with
adamc@107 228 | MFirst _ => tt
adamc@107 229 | MNext _ _ _ => tt
adamc@107 230 end
adamc@107 231 | MCons _ _ x mls' => fun mem =>
adamc@107 232 match mem in member ls' return (match ls' with
adamc@107 233 | nil => Empty_set
adamc@107 234 | x' :: ls'' =>
adamc@107 235 B x' -> (member ls'' -> B elm) -> B elm
adamc@107 236 end) with
adamc@107 237 | MFirst _ => fun x _ => x
adamc@107 238 | MNext _ _ mem' => fun _ get_mls' => get_mls' mem'
adamc@107 239 end x (hget mls')
adamc@107 240 end.
adamc@113 241 (* end thide *)
adamc@107 242 End hlist.
adamc@108 243
adamc@113 244 (* begin thide *)
adamc@108 245 Implicit Arguments MNil [A B].
adamc@108 246 Implicit Arguments MCons [A B x ls].
adamc@108 247
adamc@108 248 Implicit Arguments MFirst [A elm ls].
adamc@108 249 Implicit Arguments MNext [A elm x ls].
adamc@113 250 (* end thide *)
adamc@108 251
adamc@108 252 (** 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@108 253
adamc@108 254 Definition someTypes : list Set := nat :: bool :: nil.
adamc@108 255
adamc@113 256 (* begin thide *)
adamc@113 257
adamc@108 258 Example someValues : hlist (fun T : Set => T) someTypes :=
adamc@108 259 MCons 5 (MCons true MNil).
adamc@108 260
adamc@108 261 Eval simpl in hget someValues MFirst.
adamc@215 262 (** %\vspace{-.15in}% [[
adamc@108 263 = 5
adamc@108 264 : (fun T : Set => T) nat
adam@302 265 ]]
adam@302 266 *)
adamc@215 267
adamc@108 268 Eval simpl in hget someValues (MNext MFirst).
adamc@215 269 (** %\vspace{-.15in}% [[
adamc@108 270 = true
adamc@108 271 : (fun T : Set => T) bool
adam@302 272 ]]
adam@302 273 *)
adamc@108 274
adamc@108 275 (** We can also build indexed lists of pairs in this way. *)
adamc@108 276
adamc@108 277 Example somePairs : hlist (fun T : Set => T * T)%type someTypes :=
adamc@108 278 MCons (1, 2) (MCons (true, false) MNil).
adamc@108 279
adamc@113 280 (* end thide *)
adamc@113 281
adamc@113 282
adamc@108 283 (** ** A Lambda Calculus Interpreter *)
adamc@108 284
adam@342 285 (** 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@108 286
adamc@108 287 We start with an algebraic datatype for types. *)
adamc@108 288
adamc@108 289 Inductive type : Set :=
adamc@108 290 | Unit : type
adamc@108 291 | Arrow : type -> type -> type.
adamc@108 292
adam@342 293 (** 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@108 294
adamc@108 295 Inductive exp : list type -> type -> Set :=
adamc@108 296 | Const : forall ts, exp ts Unit
adamc@113 297 (* begin thide *)
adamc@108 298 | Var : forall ts t, member t ts -> exp ts t
adamc@108 299 | App : forall ts dom ran, exp ts (Arrow dom ran) -> exp ts dom -> exp ts ran
adamc@108 300 | Abs : forall ts dom ran, exp (dom :: ts) ran -> exp ts (Arrow dom ran).
adamc@113 301 (* end thide *)
adamc@108 302
adamc@108 303 Implicit Arguments Const [ts].
adamc@108 304
adamc@108 305 (** We write a simple recursive function to translate [type]s into [Set]s. *)
adamc@108 306
adamc@108 307 Fixpoint typeDenote (t : type) : Set :=
adamc@108 308 match t with
adamc@108 309 | Unit => unit
adamc@108 310 | Arrow t1 t2 => typeDenote t1 -> typeDenote t2
adamc@108 311 end.
adamc@108 312
adam@342 313 (** 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@108 314
adamc@113 315 (* EX: Define an interpreter for [exp]s. *)
adamc@113 316
adamc@113 317 (* begin thide *)
adamc@215 318 Fixpoint expDenote ts t (e : exp ts t) : hlist typeDenote ts -> typeDenote t :=
adamc@215 319 match e with
adamc@108 320 | Const _ => fun _ => tt
adamc@108 321
adamc@108 322 | Var _ _ mem => fun s => hget s mem
adamc@108 323 | App _ _ _ e1 e2 => fun s => (expDenote e1 s) (expDenote e2 s)
adamc@108 324 | Abs _ _ _ e' => fun s => fun x => expDenote e' (MCons x s)
adamc@108 325 end.
adamc@108 326
adamc@108 327 (** Like for previous examples, our interpreter is easy to run with [simpl]. *)
adamc@108 328
adamc@108 329 Eval simpl in expDenote Const MNil.
adamc@215 330 (** %\vspace{-.15in}% [[
adamc@108 331 = tt
adamc@108 332 : typeDenote Unit
adam@302 333 ]]
adam@302 334 *)
adamc@215 335
adamc@108 336 Eval simpl in expDenote (Abs (dom := Unit) (Var MFirst)) MNil.
adamc@215 337 (** %\vspace{-.15in}% [[
adamc@108 338 = fun x : unit => x
adamc@108 339 : typeDenote (Arrow Unit Unit)
adam@302 340 ]]
adam@302 341 *)
adamc@215 342
adamc@108 343 Eval simpl in expDenote (Abs (dom := Unit)
adamc@108 344 (Abs (dom := Unit) (Var (MNext MFirst)))) MNil.
adamc@215 345 (** %\vspace{-.15in}% [[
adamc@108 346 = fun x _ : unit => x
adamc@108 347 : typeDenote (Arrow Unit (Arrow Unit Unit))
adam@302 348 ]]
adam@302 349 *)
adamc@215 350
adamc@108 351 Eval simpl in expDenote (Abs (dom := Unit) (Abs (dom := Unit) (Var MFirst))) MNil.
adamc@215 352 (** %\vspace{-.15in}% [[
adamc@108 353 = fun _ x0 : unit => x0
adamc@108 354 : typeDenote (Arrow Unit (Arrow Unit Unit))
adam@302 355 ]]
adam@302 356 *)
adamc@215 357
adamc@108 358 Eval simpl in expDenote (App (Abs (Var MFirst)) Const) MNil.
adamc@215 359 (** %\vspace{-.15in}% [[
adamc@108 360 = tt
adamc@108 361 : typeDenote Unit
adam@302 362 ]]
adam@302 363 *)
adamc@108 364
adamc@113 365 (* end thide *)
adamc@113 366
adam@342 367 (** 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@108 368
adamc@108 369
adamc@109 370 (** * Recursive Type Definitions *)
adamc@109 371
adam@342 372 (** %\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 %\textit{%#<i>#recursive#</i>#%}% definitions. *)
adamc@109 373
adamc@113 374 (* EX: Come up with an alternate [ilist] definition that makes it easier to write [get]. *)
adamc@113 375
adamc@109 376 Section filist.
adamc@109 377 Variable A : Set.
adamc@109 378
adamc@113 379 (* begin thide *)
adamc@109 380 Fixpoint filist (n : nat) : Set :=
adamc@109 381 match n with
adamc@109 382 | O => unit
adamc@109 383 | S n' => A * filist n'
adamc@109 384 end%type.
adamc@109 385
adamc@109 386 (** 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@109 387
adamc@215 388 Fixpoint ffin (n : nat) : Set :=
adamc@109 389 match n with
adamc@109 390 | O => Empty_set
adamc@215 391 | S n' => option (ffin n')
adamc@109 392 end.
adamc@109 393
adam@342 394 (** 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@109 395
adamc@215 396 Fixpoint fget (n : nat) : filist n -> ffin n -> A :=
adamc@215 397 match n with
adamc@109 398 | O => fun _ idx => match idx with end
adamc@109 399 | S n' => fun ls idx =>
adamc@109 400 match idx with
adamc@109 401 | None => fst ls
adamc@109 402 | Some idx' => fget n' (snd ls) idx'
adamc@109 403 end
adamc@109 404 end.
adamc@109 405
adamc@215 406 (** 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@113 407 (* end thide *)
adamc@215 408
adamc@109 409 End filist.
adamc@109 410
adamc@109 411 (** Heterogeneous lists are a little trickier to define with recursion, but we then reap similar benefits in simplicity of use. *)
adamc@109 412
adamc@113 413 (* EX: Come up with an alternate [hlist] definition that makes it easier to write [hget]. *)
adamc@113 414
adamc@109 415 Section fhlist.
adamc@109 416 Variable A : Type.
adamc@109 417 Variable B : A -> Type.
adamc@109 418
adamc@113 419 (* begin thide *)
adamc@109 420 Fixpoint fhlist (ls : list A) : Type :=
adamc@109 421 match ls with
adamc@109 422 | nil => unit
adamc@109 423 | x :: ls' => B x * fhlist ls'
adamc@109 424 end%type.
adamc@109 425
adam@342 426 (** The definition of [fhlist] follows the definition of [filist], with the added wrinkle of dependently typed data elements. *)
adamc@109 427
adamc@109 428 Variable elm : A.
adamc@109 429
adamc@109 430 Fixpoint fmember (ls : list A) : Type :=
adamc@109 431 match ls with
adamc@109 432 | nil => Empty_set
adamc@109 433 | x :: ls' => (x = elm) + fmember ls'
adamc@109 434 end%type.
adamc@109 435
adamc@215 436 (** 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@109 437
adamc@109 438 We know all of the tricks needed to write a first attempt at a [get] function for [fhlist]s.
adamc@109 439
adamc@109 440 [[
adamc@109 441 Fixpoint fhget (ls : list A) : fhlist ls -> fmember ls -> B elm :=
adamc@215 442 match ls with
adamc@109 443 | nil => fun _ idx => match idx with end
adamc@109 444 | _ :: ls' => fun mls idx =>
adamc@109 445 match idx with
adamc@109 446 | inl _ => fst mls
adamc@109 447 | inr idx' => fhget ls' (snd mls) idx'
adamc@109 448 end
adamc@109 449 end.
adamc@109 450
adamc@205 451 ]]
adamc@205 452
adamc@109 453 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@109 454
adamc@109 455 Fixpoint fhget (ls : list A) : fhlist ls -> fmember ls -> B elm :=
adamc@215 456 match ls with
adamc@109 457 | nil => fun _ idx => match idx with end
adamc@109 458 | _ :: ls' => fun mls idx =>
adamc@109 459 match idx with
adamc@109 460 | inl pf => match pf with
adamc@109 461 | refl_equal => fst mls
adamc@109 462 end
adamc@109 463 | inr idx' => fhget ls' (snd mls) idx'
adamc@109 464 end
adamc@109 465 end.
adamc@109 466
adamc@109 467 (** 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@109 468
adamc@109 469 Print eq.
adamc@215 470 (** %\vspace{-.15in}% [[
adamc@109 471 Inductive eq (A : Type) (x : A) : A -> Prop := refl_equal : x = x
adamc@215 472
adamc@109 473 ]]
adamc@109 474
adamc@215 475 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@113 476 (* end thide *)
adamc@215 477
adamc@109 478 End fhlist.
adamc@110 479
adamc@111 480 Implicit Arguments fhget [A B elm ls].
adamc@111 481
adamc@110 482
adamc@110 483 (** * Data Structures as Index Functions *)
adamc@110 484
adam@342 485 (** %\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@110 486
adamc@110 487 Section tree.
adamc@110 488 Variable A : Set.
adamc@110 489
adamc@110 490 Inductive tree : Set :=
adamc@110 491 | Leaf : A -> tree
adamc@110 492 | Node : forall n, ilist tree n -> tree.
adamc@110 493 End tree.
adamc@110 494
adamc@110 495 (** 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@110 496
adamc@110 497 Section ifoldr.
adamc@110 498 Variables A B : Set.
adamc@110 499 Variable f : A -> B -> B.
adamc@110 500 Variable i : B.
adamc@110 501
adamc@215 502 Fixpoint ifoldr n (ls : ilist A n) : B :=
adamc@110 503 match ls with
adamc@110 504 | Nil => i
adamc@110 505 | Cons _ x ls' => f x (ifoldr ls')
adamc@110 506 end.
adamc@110 507 End ifoldr.
adamc@110 508
adamc@110 509 Fixpoint sum (t : tree nat) : nat :=
adamc@110 510 match t with
adamc@110 511 | Leaf n => n
adamc@110 512 | Node _ ls => ifoldr (fun t' n => sum t' + n) O ls
adamc@110 513 end.
adamc@110 514
adamc@110 515 Fixpoint inc (t : tree nat) : tree nat :=
adamc@110 516 match t with
adamc@110 517 | Leaf n => Leaf (S n)
adamc@110 518 | Node _ ls => Node (imap inc ls)
adamc@110 519 end.
adamc@110 520
adamc@110 521 (** Now we might like to prove that [inc] does not decrease a tree's [sum]. *)
adamc@110 522
adamc@110 523 Theorem sum_inc : forall t, sum (inc t) >= sum t.
adamc@113 524 (* begin thide *)
adamc@110 525 induction t; crush.
adamc@110 526 (** [[
adamc@110 527 n : nat
adamc@110 528 i : ilist (tree nat) n
adamc@110 529 ============================
adamc@110 530 ifoldr (fun (t' : tree nat) (n0 : nat) => sum t' + n0) 0 (imap inc i) >=
adamc@110 531 ifoldr (fun (t' : tree nat) (n0 : nat) => sum t' + n0) 0 i
adamc@215 532
adamc@110 533 ]]
adamc@110 534
adam@342 535 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@110 536
adamc@110 537 Check tree_ind.
adamc@215 538 (** %\vspace{-.15in}% [[
adamc@215 539 tree_ind
adamc@110 540 : forall (A : Set) (P : tree A -> Prop),
adamc@110 541 (forall a : A, P (Leaf a)) ->
adamc@110 542 (forall (n : nat) (i : ilist (tree A) n), P (Node i)) ->
adamc@110 543 forall t : tree A, P t
adamc@215 544
adamc@110 545 ]]
adamc@110 546
adam@342 547 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@215 548
adamc@110 549 Abort.
adamc@110 550
adamc@110 551 Reset tree.
adamc@110 552
adamc@110 553 (** First, let us try using our recursive definition of [ilist]s instead of the inductive version. *)
adamc@110 554
adamc@110 555 Section tree.
adamc@110 556 Variable A : Set.
adamc@110 557
adamc@215 558 (** %\vspace{-.15in}% [[
adamc@110 559 Inductive tree : Set :=
adamc@110 560 | Leaf : A -> tree
adamc@110 561 | Node : forall n, filist tree n -> tree.
adam@342 562 ]]
adamc@110 563
adam@342 564 <<
adamc@110 565 Error: Non strictly positive occurrence of "tree" in
adamc@110 566 "forall n : nat, filist tree n -> tree"
adam@342 567 >>
adamc@110 568
adam@342 569 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@110 570
adam@342 571 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 %\textit{%#<i>#define#</i>#%}% [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@110 572
adamc@110 573 Inductive tree : Set :=
adamc@110 574 | Leaf : A -> tree
adamc@215 575 | Node : forall n, (ffin n -> tree) -> tree.
adamc@110 576
adamc@215 577 (** 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@215 578
adamc@110 579 End tree.
adamc@110 580
adamc@110 581 Implicit Arguments Node [A n].
adamc@110 582
adamc@215 583 (** 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@110 584
adamc@110 585 Section rifoldr.
adamc@110 586 Variables A B : Set.
adamc@110 587 Variable f : A -> B -> B.
adamc@110 588 Variable i : B.
adamc@110 589
adamc@215 590 Fixpoint rifoldr (n : nat) : (ffin n -> A) -> B :=
adamc@215 591 match n with
adamc@110 592 | O => fun _ => i
adamc@110 593 | S n' => fun get => f (get None) (rifoldr n' (fun idx => get (Some idx)))
adamc@110 594 end.
adamc@110 595 End rifoldr.
adamc@110 596
adamc@110 597 Implicit Arguments rifoldr [A B n].
adamc@110 598
adamc@110 599 Fixpoint sum (t : tree nat) : nat :=
adamc@110 600 match t with
adamc@110 601 | Leaf n => n
adamc@110 602 | Node _ f => rifoldr plus O (fun idx => sum (f idx))
adamc@110 603 end.
adamc@110 604
adamc@110 605 Fixpoint inc (t : tree nat) : tree nat :=
adamc@110 606 match t with
adamc@110 607 | Leaf n => Leaf (S n)
adamc@110 608 | Node _ f => Node (fun idx => inc (f idx))
adamc@110 609 end.
adamc@110 610
adamc@110 611 (** 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 %\textit{%#<i>#will#</i>#%}% be helpful to prove some lemmas. *)
adamc@110 612
adamc@110 613 Lemma plus_ge : forall x1 y1 x2 y2,
adamc@110 614 x1 >= x2
adamc@110 615 -> y1 >= y2
adamc@110 616 -> x1 + y1 >= x2 + y2.
adamc@110 617 crush.
adamc@110 618 Qed.
adamc@110 619
adamc@215 620 Lemma sum_inc' : forall n (f1 f2 : ffin n -> nat),
adamc@110 621 (forall idx, f1 idx >= f2 idx)
adamc@110 622 -> rifoldr plus 0 f1 >= rifoldr plus 0 f2.
adamc@110 623 Hint Resolve plus_ge.
adamc@110 624
adamc@110 625 induction n; crush.
adamc@110 626 Qed.
adamc@110 627
adamc@110 628 Theorem sum_inc : forall t, sum (inc t) >= sum t.
adamc@110 629 Hint Resolve sum_inc'.
adamc@110 630
adamc@110 631 induction t; crush.
adamc@110 632 Qed.
adamc@110 633
adamc@113 634 (* end thide *)
adamc@113 635
adamc@110 636 (** 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@111 637
adamc@111 638 (** ** Another Interpreter Example *)
adamc@111 639
adam@342 640 (** 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@112 641
adamc@112 642 Inductive type' : Type := Nat | Bool.
adamc@111 643
adamc@111 644 Inductive exp' : type' -> Type :=
adamc@112 645 | NConst : nat -> exp' Nat
adamc@112 646 | Plus : exp' Nat -> exp' Nat -> exp' Nat
adamc@112 647 | Eq : exp' Nat -> exp' Nat -> exp' Bool
adamc@111 648
adamc@112 649 | BConst : bool -> exp' Bool
adamc@113 650 (* begin thide *)
adamc@215 651 | Cond : forall n t, (ffin n -> exp' Bool)
adamc@215 652 -> (ffin n -> exp' t) -> exp' t -> exp' t.
adamc@113 653 (* end thide *)
adamc@111 654
adam@284 655 (** 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@112 656
adam@284 657 Example ex1 := Cond 2
adam@284 658 (fun f => match f with
adam@284 659 | None => Eq (Plus (NConst 2) (NConst 2)) (NConst 5)
adam@284 660 | Some None => Eq (Plus (NConst 1) (NConst 1)) (NConst 2)
adam@284 661 | Some (Some v) => match v with end
adam@284 662 end)
adam@284 663 (fun f => match f with
adam@284 664 | None => NConst 0
adam@284 665 | Some None => NConst 1
adam@284 666 | Some (Some v) => match v with end
adam@284 667 end)
adam@284 668 (NConst 2).
adam@284 669
adam@284 670 (** We start implementing our interpreter with a standard type denotation function. *)
adamc@112 671
adamc@111 672 Definition type'Denote (t : type') : Set :=
adamc@111 673 match t with
adamc@112 674 | Nat => nat
adamc@112 675 | Bool => bool
adamc@111 676 end.
adamc@111 677
adamc@112 678 (** To implement the expression interpreter, it is useful to have the following function that implements the functionality of [Cond] without involving any syntax. *)
adamc@112 679
adamc@113 680 (* begin thide *)
adamc@111 681 Section cond.
adamc@111 682 Variable A : Set.
adamc@111 683 Variable default : A.
adamc@111 684
adamc@215 685 Fixpoint cond (n : nat) : (ffin n -> bool) -> (ffin n -> A) -> A :=
adamc@215 686 match n with
adamc@111 687 | O => fun _ _ => default
adamc@111 688 | S n' => fun tests bodies =>
adamc@111 689 if tests None
adamc@111 690 then bodies None
adamc@111 691 else cond n'
adamc@111 692 (fun idx => tests (Some idx))
adamc@111 693 (fun idx => bodies (Some idx))
adamc@111 694 end.
adamc@111 695 End cond.
adamc@111 696
adamc@111 697 Implicit Arguments cond [A n].
adamc@113 698 (* end thide *)
adamc@111 699
adamc@112 700 (** Now the expression interpreter is straightforward to write. *)
adamc@112 701
adamc@215 702 Fixpoint exp'Denote t (e : exp' t) : type'Denote t :=
adamc@215 703 match e with
adamc@215 704 | NConst n => n
adamc@215 705 | Plus e1 e2 => exp'Denote e1 + exp'Denote e2
adamc@111 706 | Eq e1 e2 =>
adamc@111 707 if eq_nat_dec (exp'Denote e1) (exp'Denote e2) then true else false
adamc@111 708
adamc@215 709 | BConst b => b
adamc@111 710 | Cond _ _ tests bodies default =>
adamc@113 711 (* begin thide *)
adamc@111 712 cond
adamc@111 713 (exp'Denote default)
adamc@111 714 (fun idx => exp'Denote (tests idx))
adamc@111 715 (fun idx => exp'Denote (bodies idx))
adamc@113 716 (* end thide *)
adamc@111 717 end.
adamc@111 718
adamc@112 719 (** 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@112 720
adamc@113 721 (* begin thide *)
adamc@111 722 Section cfoldCond.
adamc@111 723 Variable t : type'.
adamc@111 724 Variable default : exp' t.
adamc@111 725
adamc@112 726 Fixpoint cfoldCond (n : nat)
adamc@215 727 : (ffin n -> exp' Bool) -> (ffin n -> exp' t) -> exp' t :=
adamc@215 728 match n with
adamc@111 729 | O => fun _ _ => default
adamc@111 730 | S n' => fun tests bodies =>
adamc@204 731 match tests None return _ with
adamc@111 732 | BConst true => bodies None
adamc@111 733 | BConst false => cfoldCond n'
adamc@111 734 (fun idx => tests (Some idx))
adamc@111 735 (fun idx => bodies (Some idx))
adamc@111 736 | _ =>
adamc@111 737 let e := cfoldCond n'
adamc@111 738 (fun idx => tests (Some idx))
adamc@111 739 (fun idx => bodies (Some idx)) in
adamc@112 740 match e in exp' t return exp' t -> exp' t with
adamc@112 741 | Cond n _ tests' bodies' default' => fun body =>
adamc@111 742 Cond
adamc@111 743 (S n)
adamc@111 744 (fun idx => match idx with
adamc@112 745 | None => tests None
adamc@111 746 | Some idx => tests' idx
adamc@111 747 end)
adamc@111 748 (fun idx => match idx with
adamc@111 749 | None => body
adamc@111 750 | Some idx => bodies' idx
adamc@111 751 end)
adamc@111 752 default'
adamc@112 753 | e => fun body =>
adamc@111 754 Cond
adamc@111 755 1
adamc@112 756 (fun _ => tests None)
adamc@111 757 (fun _ => body)
adamc@111 758 e
adamc@112 759 end (bodies None)
adamc@111 760 end
adamc@111 761 end.
adamc@111 762 End cfoldCond.
adamc@111 763
adamc@111 764 Implicit Arguments cfoldCond [t n].
adamc@113 765 (* end thide *)
adamc@111 766
adamc@112 767 (** Like for the interpreters, most of the action was in this helper function, and [cfold] itself is easy to write. *)
adamc@112 768
adamc@215 769 Fixpoint cfold t (e : exp' t) : exp' t :=
adamc@215 770 match e with
adamc@111 771 | NConst n => NConst n
adamc@111 772 | Plus e1 e2 =>
adamc@111 773 let e1' := cfold e1 in
adamc@111 774 let e2' := cfold e2 in
adamc@204 775 match e1', e2' return _ with
adamc@111 776 | NConst n1, NConst n2 => NConst (n1 + n2)
adamc@111 777 | _, _ => Plus e1' e2'
adamc@111 778 end
adamc@111 779 | Eq e1 e2 =>
adamc@111 780 let e1' := cfold e1 in
adamc@111 781 let e2' := cfold e2 in
adamc@204 782 match e1', e2' return _ with
adamc@111 783 | NConst n1, NConst n2 => BConst (if eq_nat_dec n1 n2 then true else false)
adamc@111 784 | _, _ => Eq e1' e2'
adamc@111 785 end
adamc@111 786
adamc@111 787 | BConst b => BConst b
adamc@111 788 | Cond _ _ tests bodies default =>
adamc@113 789 (* begin thide *)
adamc@111 790 cfoldCond
adamc@111 791 (cfold default)
adamc@111 792 (fun idx => cfold (tests idx))
adamc@111 793 (fun idx => cfold (bodies idx))
adamc@113 794 (* end thide *)
adamc@111 795 end.
adamc@111 796
adamc@113 797 (* begin thide *)
adam@342 798 (** 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@112 799
adamc@111 800 Lemma cfoldCond_correct : forall t (default : exp' t)
adamc@215 801 n (tests : ffin n -> exp' Bool) (bodies : ffin n -> exp' t),
adamc@111 802 exp'Denote (cfoldCond default tests bodies)
adamc@111 803 = exp'Denote (Cond n tests bodies default).
adamc@111 804 induction n; crush;
adamc@111 805 match goal with
adamc@111 806 | [ IHn : forall tests bodies, _, tests : _ -> _, bodies : _ -> _ |- _ ] =>
adam@294 807 specialize (IHn (fun idx => tests (Some idx)) (fun idx => bodies (Some idx)))
adamc@111 808 end;
adamc@111 809 repeat (match goal with
adamc@111 810 | [ |- context[match ?E with
adamc@111 811 | NConst _ => _
adamc@111 812 | Plus _ _ => _
adamc@111 813 | Eq _ _ => _
adamc@111 814 | BConst _ => _
adamc@111 815 | Cond _ _ _ _ _ => _
adamc@111 816 end] ] => dep_destruct E
adamc@111 817 | [ |- context[if ?B then _ else _] ] => destruct B
adamc@111 818 end; crush).
adamc@111 819 Qed.
adamc@111 820
adamc@112 821 (** 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 %\textit{%#<i>#extensional#</i>#%}%; that is, it is unaffected by substitution of functions with input-output equivalents. *)
adamc@112 822
adamc@215 823 Lemma cond_ext : forall (A : Set) (default : A) n (tests tests' : ffin n -> bool)
adamc@215 824 (bodies bodies' : ffin n -> A),
adamc@111 825 (forall idx, tests idx = tests' idx)
adamc@111 826 -> (forall idx, bodies idx = bodies' idx)
adamc@111 827 -> cond default tests bodies
adamc@111 828 = cond default tests' bodies'.
adamc@111 829 induction n; crush;
adamc@111 830 match goal with
adamc@111 831 | [ |- context[if ?E then _ else _] ] => destruct E
adamc@111 832 end; crush.
adamc@111 833 Qed.
adamc@111 834
adamc@112 835 (** 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@113 836 (* end thide *)
adamc@112 837
adamc@111 838 Theorem cfold_correct : forall t (e : exp' t),
adamc@111 839 exp'Denote (cfold e) = exp'Denote e.
adamc@113 840 (* begin thide *)
adamc@111 841 Hint Rewrite cfoldCond_correct : cpdt.
adamc@111 842 Hint Resolve cond_ext.
adamc@111 843
adamc@111 844 induction e; crush;
adamc@111 845 repeat (match goal with
adamc@111 846 | [ |- context[cfold ?E] ] => dep_destruct (cfold E)
adamc@111 847 end; crush).
adamc@111 848 Qed.
adamc@113 849 (* end thide *)
adamc@115 850
adamc@115 851
adamc@215 852 (** * Choosing Between Representations *)
adamc@215 853
adamc@215 854 (** 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@215 855
adamc@215 856 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@215 857
adam@342 858 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@215 859
adam@342 860 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@215 861
adam@342 862 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@342 863
adam@342 864 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. *)
adamc@215 865
adamc@215 866
adamc@115 867 (** * Exercises *)
adamc@115 868
adamc@116 869 (** remove printing * *)
adamc@116 870
adamc@216 871 (** Some of the type family definitions and associated functions from this chapter are duplicated in the [DepList] module of the book source. Some of their names have been changed to be more sensible in a general context.
adamc@115 872
adamc@115 873 %\begin{enumerate}%#<ol>#
adamc@115 874
adam@284 875 %\item%#<li># Define a tree analogue of [hlist]. That is, define a parameterized type of binary trees with data at their leaves, and define a type family [htree] indexed by trees. The structure of an [htree] mirrors its index tree, with the type of each data element (which only occur at leaves) determined by applying a type function to the corresponding element of the index tree. Define a type standing for all possible paths from the root of a tree to leaves and use it to implement a function [tget] for extracting an element of an [htree] by path. Define a function [htmap2] for %``%#"#mapping over two trees in parallel.#"#%''% That is, [htmap2] takes in two [htree]s with the same index tree, and it forms a new [htree] with the same index by applying a binary function pointwise.
adamc@115 876
adamc@129 877 Repeat this process so that you implement each definition for each of the three definition styles covered in this chapter: inductive, recursive, and index function.#</li>#
adamc@116 878
adam@342 879 %\item%#<li># Write a dependently typed interpreter for a simple programming language with ML-style pattern-matching, using one of the encodings of heterogeneous lists to represent the different branches of a [case] expression. (There are other ways to represent the same thing, but the point of this exercise is to practice using those heterogeneous list types.) The object language is defined informally by this grammar:
adamc@116 880 [[
adamc@116 881 t ::= bool | t + t
adamc@116 882 p ::= x | b | inl p | inr p
adamc@116 883 e ::= x | b | inl e | inr e | case e of [p => e]* | _ => e
adamc@116 884 ]]
adamc@116 885
adam@342 886 The non-terminal [x] stands for a variable, and [b] stands for a boolean constant. The production for [case] expressions means that a pattern-match includes zero or more pairs of patterns and expressions, along with a default case.
adamc@116 887
adamc@117 888 Your interpreter should be implemented in the style demonstrated in this chapter. That is, your definition of expressions should use dependent types and de Bruijn indices to combine syntax and typing rules, such that the type of an expression tells the types of variables that are in scope. You should implement a simple recursive function translating types [t] to [Set], and your interpreter should produce values in the image of this translation.#</li>#
adamc@116 889
adamc@115 890 #</ol>#%\end{enumerate}% *)