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Citations for lazy data structures, Haskell 'deriving', and logic programming
author Adam Chlipala <adam@chlipala.net>
date Sun, 22 Apr 2012 16:25:22 -0400
parents d1276004eec9
children 05efde66559d
rev   line source
adam@286 1 (* Copyright (c) 2008-2010, Adam Chlipala
adamc@193 2 *
adamc@193 3 * This work is licensed under a
adamc@193 4 * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
adamc@193 5 * Unported License.
adamc@193 6 * The license text is available at:
adamc@193 7 * http://creativecommons.org/licenses/by-nc-nd/3.0/
adamc@193 8 *)
adamc@193 9
adamc@193 10 (* begin hide *)
adamc@195 11 Require Import String List.
adamc@193 12
adam@314 13 Require Import CpdtTactics DepList.
adamc@193 14
adamc@193 15 Set Implicit Arguments.
adamc@193 16 (* end hide *)
adamc@193 17
adamc@193 18
adamc@219 19 (** %\chapter{Generic Programming}% *)
adamc@193 20
adam@395 21 (** %\index{generic programming}\textit{%#<i>#Generic programming#</i>#%}% makes it possible to write functions that operate over different types of data. %\index{parametric polymorphism}%Parametric polymorphism in ML and Haskell is one of the simplest examples. ML-style %\index{module systems}%module systems%~\cite{modules}% and Haskell %\index{type classes}%type classes%~\cite{typeclasses}% are more flexible cases. These language features are often not as powerful as we would like. For instance, while Haskell includes a type class classifying those types whose values can be pretty-printed, per-type pretty-printing is usually either implemented manually or implemented via a %\index{deriving clauses}%[deriving] clause%~\cite{deriving}%, which triggers ad-hoc code generation. Some clever encoding tricks have been used to achieve better within Haskell and other languages, but we can do %\index{datatype-generic programming}\emph{%#<i>#datatype-generic programming#</i>#%}% much more cleanly with dependent types. Thanks to the expressive power of CIC, we need no special language support.
adamc@193 22
adamc@219 23 Generic programming can often be very useful in Coq developments, so we devote this chapter to studying it. In a proof assistant, there is the new possibility of generic proofs about generic programs, which we also devote some space to. *)
adamc@193 24
adamc@195 25 (** * Reflecting Datatype Definitions *)
adamc@193 26
adam@358 27 (** The key to generic programming with dependent types is %\index{universe types}\textit{%#<i>#universe types#</i>#%}%. This concept should not be confused with the idea of %\textit{%#<i>#universes#</i>#%}% from the metatheory of CIC and related languages. Rather, the idea of universe types is to define inductive types that provide %\textit{%#<i>#syntactic representations#</i>#%}% of Coq types. We cannot directly write CIC programs that do case analysis on types, but we %\textit{%#<i>#can#</i>#%}% case analyze on reflected syntactic versions of those types.
adamc@219 28
adam@358 29 Thus, to begin, we must define a syntactic representation of some class of datatypes. In this chapter, our running example will have to do with basic algebraic datatypes, of the kind found in ML and Haskell, but without additional bells and whistles like type parameters and mutually recursive definitions.
adamc@219 30
adamc@219 31 The first step is to define a representation for constructors of our datatypes. *)
adamc@219 32
adamc@198 33 (* EX: Define a reflected representation of simple algebraic datatypes. *)
adamc@198 34
adamc@198 35 (* begin thide *)
adamc@193 36 Record constructor : Type := Con {
adamc@193 37 nonrecursive : Type;
adamc@193 38 recursive : nat
adamc@193 39 }.
adamc@193 40
adam@286 41 (** The idea is that a constructor represented as [Con T n] has [n] arguments of the type that we are defining. Additionally, all of the other, non-recursive arguments can be encoded in the type [T]. When there are no non-recursive arguments, [T] can be [unit]. When there are two non-recursive arguments, of types [A] and [B], [T] can be [A * B]. We can generalize to any number of arguments via tupling.
adamc@219 42
adamc@219 43 With this definition, it as easy to define a datatype representation in terms of lists of constructors. *)
adamc@219 44
adamc@193 45 Definition datatype := list constructor.
adamc@193 46
adamc@219 47 (** Here are a few example encodings for some common types from the Coq standard library. While our syntax type does not support type parameters directly, we can implement them at the meta level, via functions from types to [datatype]s. *)
adamc@219 48
adamc@193 49 Definition Empty_set_dt : datatype := nil.
adamc@193 50 Definition unit_dt : datatype := Con unit 0 :: nil.
adamc@193 51 Definition bool_dt : datatype := Con unit 0 :: Con unit 0 :: nil.
adamc@193 52 Definition nat_dt : datatype := Con unit 0 :: Con unit 1 :: nil.
adamc@193 53 Definition list_dt (A : Type) : datatype := Con unit 0 :: Con A 1 :: nil.
adamc@219 54
adam@358 55 (** The type [Empty_set] has no constructors, so its representation is the empty list. The type [unit] has one constructor with no arguments, so its one reflected constructor indicates no non-recursive data and [0] recursive arguments. The representation for [bool] just duplicates this single argumentless constructor. We get from [bool] to [nat] by changing one of the constructors to indicate 1 recursive argument. We get from [nat] to [list] by adding a non-recursive argument of a parameter type [A].
adamc@219 56
adamc@219 57 As a further example, we can do the same encoding for a generic binary tree type. *)
adamc@219 58
adamc@198 59 (* end thide *)
adamc@193 60
adamc@193 61 Section tree.
adamc@193 62 Variable A : Type.
adamc@193 63
adamc@193 64 Inductive tree : Type :=
adamc@193 65 | Leaf : A -> tree
adamc@193 66 | Node : tree -> tree -> tree.
adamc@193 67 End tree.
adamc@193 68
adamc@198 69 (* begin thide *)
adamc@193 70 Definition tree_dt (A : Type) : datatype := Con A 0 :: Con unit 2 :: nil.
adamc@193 71
adamc@219 72 (** Each datatype representation stands for a family of inductive types. For a specific real datatype and a reputed representation for it, it is useful to define a type of %\textit{%#<i>#evidence#</i>#%}% that the datatype is compatible with the encoding. *)
adamc@219 73
adamc@193 74 Section denote.
adamc@193 75 Variable T : Type.
adamc@219 76 (** This variable stands for the concrete datatype that we are interested in. *)
adamc@193 77
adamc@193 78 Definition constructorDenote (c : constructor) :=
adamc@193 79 nonrecursive c -> ilist T (recursive c) -> T.
adam@358 80 (** We write that a constructor is represented as a function returning a [T]. Such a function takes two arguments, which pack together the non-recursive and recursive arguments of the constructor. We represent a tuple of all recursive arguments using the length-indexed list type %\index{Gallina terms!ilist}%[ilist] that we met in Chapter 8. *)
adamc@193 81
adamc@193 82 Definition datatypeDenote := hlist constructorDenote.
adam@358 83 (** Finally, the evidence for type [T] is a %\index{Gallina terms!hlist}%heterogeneous list, including a constructor denotation for every constructor encoding in a datatype encoding. Recall that, since we are inside a section binding [T] as a variable, [constructorDenote] is automatically parameterized by [T]. *)
adamc@219 84
adamc@193 85 End denote.
adamc@198 86 (* end thide *)
adamc@193 87
adamc@219 88 (** Some example pieces of evidence should help clarify the convention. First, we define some helpful notations, providing different ways of writing constructor denotations. There is really just one notation, but we need several versions of it to cover different choices of which variables will be used in the body of a definition. %The ASCII \texttt{\textasciitilde{}>} from the notation will be rendered later as $\leadsto$.% *)
adamc@219 89
adamc@219 90 (** printing ~> $\leadsto$ *)
adamc@219 91
adamc@193 92 Notation "[ ! , ! ~> x ]" := ((fun _ _ => x) : constructorDenote _ (Con _ _)).
adamc@193 93 Notation "[ v , ! ~> x ]" := ((fun v _ => x) : constructorDenote _ (Con _ _)).
adamc@219 94 Notation "[ ! , r ~> x ]" := ((fun _ r => x) : constructorDenote _ (Con _ _)).
adamc@219 95 Notation "[ v , r ~> x ]" := ((fun v r => x) : constructorDenote _ (Con _ _)).
adamc@193 96
adamc@198 97 (* begin thide *)
adamc@193 98 Definition Empty_set_den : datatypeDenote Empty_set Empty_set_dt :=
adamc@216 99 HNil.
adamc@193 100 Definition unit_den : datatypeDenote unit unit_dt :=
adamc@216 101 [!, ! ~> tt] ::: HNil.
adamc@193 102 Definition bool_den : datatypeDenote bool bool_dt :=
adamc@216 103 [!, ! ~> true] ::: [!, ! ~> false] ::: HNil.
adamc@193 104 Definition nat_den : datatypeDenote nat nat_dt :=
adamc@219 105 [!, ! ~> O] ::: [!, r ~> S (hd r)] ::: HNil.
adamc@193 106 Definition list_den (A : Type) : datatypeDenote (list A) (list_dt A) :=
adamc@219 107 [!, ! ~> nil] ::: [x, r ~> x :: hd r] ::: HNil.
adamc@193 108 Definition tree_den (A : Type) : datatypeDenote (tree A) (tree_dt A) :=
adamc@219 109 [v, ! ~> Leaf v] ::: [!, r ~> Node (hd r) (hd (tl r))] ::: HNil.
adamc@198 110 (* end thide *)
adamc@194 111
adam@358 112 (** Recall that the [hd] and [tl] calls above operate on richly typed lists, where type indices tell us the lengths of lists, guaranteeing the safety of operations like [hd]. The type annotation attached to each definition provides enough information for Coq to infer list lengths at appropriate points. *)
adam@358 113
adamc@195 114
adamc@195 115 (** * Recursive Definitions *)
adamc@195 116
adamc@198 117 (* EX: Define a generic [size] function. *)
adamc@198 118
adam@358 119 (** We built these encodings of datatypes to help us write datatype-generic recursive functions. To do so, we will want a reflected representation of a %\index{recursion schemes}\textit{%#<i>#recursion scheme#</i>#%}% for each type, similar to the [T_rect] principle generated automatically for an inductive definition of [T]. A clever reuse of [datatypeDenote] yields a short definition. *)
adamc@219 120
adamc@198 121 (* begin thide *)
adamc@194 122 Definition fixDenote (T : Type) (dt : datatype) :=
adamc@194 123 forall (R : Type), datatypeDenote R dt -> (T -> R).
adamc@194 124
adamc@219 125 (** The idea of a recursion scheme is parameterized by a type and a reputed encoding of it. The principle itself is polymorphic in a type [R], which is the return type of the recursive function that we mean to write. The next argument is a hetergeneous list of one case of the recursive function definition for each datatype constructor. The [datatypeDenote] function turns out to have just the right definition to express the type we need; a set of function cases is just like an alternate set of constructors where we replace the original type [T] with the function result type [R]. Given such a reflected definition, a [fixDenote] invocation returns a function from [T] to [R], which is just what we wanted.
adamc@219 126
adamc@219 127 We are ready to write some example functions now. It will be useful to use one new function from the [DepList] library included in the book source. *)
adamc@219 128
adamc@219 129 Check hmake.
adamc@219 130 (** %\vspace{-.15in}% [[
adamc@219 131 hmake
adamc@219 132 : forall (A : Type) (B : A -> Type),
adam@358 133 (forall x : A, B x) -> forall ls : list A, hlist B ls
adamc@219 134 ]]
adamc@219 135
adam@358 136 The function [hmake] is a kind of [map] alternative that goes from a regular [list] to an [hlist]. We can use it to define a generic size function that counts the number of constructors used to build a value in a datatype. *)
adamc@219 137
adamc@194 138 Definition size T dt (fx : fixDenote T dt) : T -> nat :=
adamc@194 139 fx nat (hmake (B := constructorDenote nat) (fun _ _ r => foldr plus 1 r) dt).
adamc@194 140
adamc@219 141 (** Our definition is parameterized over a recursion scheme [fx]. We instantiate [fx] by passing it the function result type and a set of function cases, where we build the latter with [hmake]. The function argument to [hmake] takes three arguments: the representation of a constructor, its non-recursive arguments, and the results of recursive calls on all of its recursive arguments. We only need the recursive call results here, so we call them [r] and bind the other two inputs with wildcards. The actual case body is simple: we add together the recursive call results and increment the result by one (to account for the current constructor). This [foldr] function is an [hlist]-specific version defined in the [DepList] module.
adamc@219 142
adamc@219 143 It is instructive to build [fixDenote] values for our example types and see what specialized [size] functions result from them. *)
adamc@219 144
adamc@194 145 Definition Empty_set_fix : fixDenote Empty_set Empty_set_dt :=
adamc@194 146 fun R _ emp => match emp with end.
adamc@194 147 Eval compute in size Empty_set_fix.
adamc@219 148 (** %\vspace{-.15in}% [[
adamc@219 149 = fun emp : Empty_set => match emp return nat with
adamc@219 150 end
adamc@219 151 : Empty_set -> nat
adamc@219 152 ]]
adamc@219 153
adamc@219 154 Despite all the fanciness of the generic [size] function, CIC's standard computation rules suffice to normalize the generic function specialization to exactly what we would have written manually. *)
adamc@194 155
adamc@194 156 Definition unit_fix : fixDenote unit unit_dt :=
adamc@216 157 fun R cases _ => (hhd cases) tt INil.
adamc@194 158 Eval compute in size unit_fix.
adamc@219 159 (** %\vspace{-.15in}% [[
adamc@219 160 = fun _ : unit => 1
adamc@219 161 : unit -> nat
adamc@219 162 ]]
adamc@219 163
adamc@219 164 Again normalization gives us the natural function definition. We see this pattern repeated for our other example types. *)
adamc@194 165
adamc@194 166 Definition bool_fix : fixDenote bool bool_dt :=
adamc@194 167 fun R cases b => if b
adamc@216 168 then (hhd cases) tt INil
adamc@216 169 else (hhd (htl cases)) tt INil.
adamc@194 170 Eval compute in size bool_fix.
adamc@219 171 (** %\vspace{-.15in}% [[
adamc@219 172 = fun b : bool => if b then 1 else 1
adamc@219 173 : bool -> nat
adam@302 174 ]]
adam@302 175 *)
adamc@194 176
adamc@194 177 Definition nat_fix : fixDenote nat nat_dt :=
adamc@194 178 fun R cases => fix F (n : nat) : R :=
adamc@194 179 match n with
adamc@216 180 | O => (hhd cases) tt INil
adamc@216 181 | S n' => (hhd (htl cases)) tt (ICons (F n') INil)
adamc@194 182 end.
adamc@219 183
adamc@219 184 (** To peek at the [size] function for [nat], it is useful to avoid full computation, so that the recursive definition of addition is not expanded inline. We can accomplish this with proper flags for the [cbv] reduction strategy. *)
adamc@219 185
adamc@194 186 Eval cbv beta iota delta -[plus] in size nat_fix.
adamc@219 187 (** %\vspace{-.15in}% [[
adamc@219 188 = fix F (n : nat) : nat := match n with
adamc@219 189 | 0 => 1
adamc@219 190 | S n' => F n' + 1
adamc@219 191 end
adamc@219 192 : nat -> nat
adam@302 193 ]]
adam@302 194 *)
adamc@194 195
adamc@194 196 Definition list_fix (A : Type) : fixDenote (list A) (list_dt A) :=
adamc@194 197 fun R cases => fix F (ls : list A) : R :=
adamc@194 198 match ls with
adamc@216 199 | nil => (hhd cases) tt INil
adamc@216 200 | x :: ls' => (hhd (htl cases)) x (ICons (F ls') INil)
adamc@194 201 end.
adamc@194 202 Eval cbv beta iota delta -[plus] in fun A => size (@list_fix A).
adamc@219 203 (** %\vspace{-.15in}% [[
adamc@219 204 = fun A : Type =>
adamc@219 205 fix F (ls : list A) : nat :=
adamc@219 206 match ls with
adamc@219 207 | nil => 1
adamc@219 208 | _ :: ls' => F ls' + 1
adamc@219 209 end
adamc@219 210 : forall A : Type, list A -> nat
adam@302 211 ]]
adam@302 212 *)
adamc@194 213
adamc@194 214 Definition tree_fix (A : Type) : fixDenote (tree A) (tree_dt A) :=
adamc@194 215 fun R cases => fix F (t : tree A) : R :=
adamc@194 216 match t with
adamc@216 217 | Leaf x => (hhd cases) x INil
adamc@216 218 | Node t1 t2 => (hhd (htl cases)) tt (ICons (F t1) (ICons (F t2) INil))
adamc@194 219 end.
adamc@194 220 Eval cbv beta iota delta -[plus] in fun A => size (@tree_fix A).
adamc@219 221 (** %\vspace{-.15in}% [[
adamc@219 222 = fun A : Type =>
adamc@219 223 fix F (t : tree A) : nat :=
adamc@219 224 match t with
adamc@219 225 | Leaf _ => 1
adamc@219 226 | Node t1 t2 => F t1 + (F t2 + 1)
adamc@219 227 end
adamc@219 228 : forall A : Type, tree A -> n
adam@302 229 ]]
adam@302 230 *)
adamc@198 231 (* end thide *)
adamc@195 232
adamc@195 233
adamc@195 234 (** ** Pretty-Printing *)
adamc@195 235
adamc@219 236 (** It is also useful to do generic pretty-printing of datatype values, rendering them as human-readable strings. To do so, we will need a bit of metadata for each constructor. Specifically, we need the name to print for the constructor and the function to use to render its non-recursive arguments. Everything else can be done generically. *)
adamc@219 237
adamc@195 238 Record print_constructor (c : constructor) : Type := PI {
adamc@195 239 printName : string;
adamc@195 240 printNonrec : nonrecursive c -> string
adamc@195 241 }.
adamc@195 242
adamc@219 243 (** It is useful to define a shorthand for applying the constructor [PI]. By applying it explicitly to an unknown application of the constructor [Con], we help type inference work. *)
adamc@219 244
adamc@195 245 Notation "^" := (PI (Con _ _)).
adamc@195 246
adamc@219 247 (** As in earlier examples, we define the type of metadata for a datatype to be a heterogeneous list type collecting metadata for each constructor. *)
adamc@219 248
adamc@195 249 Definition print_datatype := hlist print_constructor.
adamc@195 250
adamc@219 251 (** We will be doing some string manipulation here, so we import the notations associated with strings. *)
adamc@219 252
adamc@219 253 Local Open Scope string_scope.
adamc@219 254
adamc@219 255 (** Now it is easy to implement our generic printer, using another function from [DepList.] *)
adamc@219 256
adamc@219 257 Check hmap.
adamc@219 258 (** %\vspace{-.15in}% [[
adamc@219 259 hmap
adamc@219 260 : forall (A : Type) (B1 B2 : A -> Type),
adamc@219 261 (forall x : A, B1 x -> B2 x) ->
adamc@219 262 forall ls : list A, hlist B1 ls -> hlist B2 ls
adam@302 263 ]]
adam@302 264 *)
adamc@195 265
adamc@195 266 Definition print T dt (pr : print_datatype dt) (fx : fixDenote T dt) : T -> string :=
adamc@195 267 fx string (hmap (B1 := print_constructor) (B2 := constructorDenote string)
adamc@195 268 (fun _ pc x r => printName pc ++ "(" ++ printNonrec pc x
adamc@195 269 ++ foldr (fun s acc => ", " ++ s ++ acc) ")" r) pr).
adamc@195 270
adamc@219 271 (** Some simple tests establish that [print] gets the job done. *)
adamc@219 272
adamc@216 273 Eval compute in print HNil Empty_set_fix.
adamc@219 274 (** %\vspace{-.15in}% [[
adamc@219 275 = fun emp : Empty_set => match emp return string with
adamc@219 276 end
adamc@219 277 : Empty_set -> string
adam@302 278 ]]
adam@302 279 *)
adamc@219 280
adamc@216 281 Eval compute in print (^ "tt" (fun _ => "") ::: HNil) unit_fix.
adamc@219 282 (** %\vspace{-.15in}% [[
adamc@219 283 = fun _ : unit => "tt()"
adamc@219 284 : unit -> string
adam@302 285 ]]
adam@302 286 *)
adamc@219 287
adamc@195 288 Eval compute in print (^ "true" (fun _ => "")
adamc@195 289 ::: ^ "false" (fun _ => "")
adamc@216 290 ::: HNil) bool_fix.
adamc@219 291 (** %\vspace{-.15in}% [[
adamc@219 292 = fun b : bool => if b then "true()" else "false()"
adamc@219 293 : bool -> s
adam@302 294 ]]
adam@302 295 *)
adamc@195 296
adamc@195 297 Definition print_nat := print (^ "O" (fun _ => "")
adamc@195 298 ::: ^ "S" (fun _ => "")
adamc@216 299 ::: HNil) nat_fix.
adamc@195 300 Eval cbv beta iota delta -[append] in print_nat.
adamc@219 301 (** %\vspace{-.15in}% [[
adamc@219 302 = fix F (n : nat) : string :=
adamc@219 303 match n with
adamc@219 304 | 0%nat => "O" ++ "(" ++ "" ++ ")"
adamc@219 305 | S n' => "S" ++ "(" ++ "" ++ ", " ++ F n' ++ ")"
adamc@219 306 end
adamc@219 307 : nat -> string
adam@302 308 ]]
adam@302 309 *)
adamc@219 310
adamc@195 311 Eval simpl in print_nat 0.
adamc@219 312 (** %\vspace{-.15in}% [[
adamc@219 313 = "O()"
adamc@219 314 : string
adam@302 315 ]]
adam@302 316 *)
adamc@219 317
adamc@195 318 Eval simpl in print_nat 1.
adamc@219 319 (** %\vspace{-.15in}% [[
adamc@219 320 = "S(, O())"
adamc@219 321 : string
adam@302 322 ]]
adam@302 323 *)
adamc@219 324
adamc@195 325 Eval simpl in print_nat 2.
adamc@219 326 (** %\vspace{-.15in}% [[
adamc@219 327 = "S(, S(, O()))"
adamc@219 328 : string
adam@302 329 ]]
adam@302 330 *)
adamc@195 331
adamc@195 332 Eval cbv beta iota delta -[append] in fun A (pr : A -> string) =>
adamc@195 333 print (^ "nil" (fun _ => "")
adamc@195 334 ::: ^ "cons" pr
adamc@216 335 ::: HNil) (@list_fix A).
adamc@219 336 (** %\vspace{-.15in}% [[
adamc@219 337 = fun (A : Type) (pr : A -> string) =>
adamc@219 338 fix F (ls : list A) : string :=
adamc@219 339 match ls with
adamc@219 340 | nil => "nil" ++ "(" ++ "" ++ ")"
adamc@219 341 | x :: ls' => "cons" ++ "(" ++ pr x ++ ", " ++ F ls' ++ ")"
adamc@219 342 end
adamc@219 343 : forall A : Type, (A -> string) -> list A -> string
adam@302 344 ]]
adam@302 345 *)
adamc@219 346
adamc@195 347 Eval cbv beta iota delta -[append] in fun A (pr : A -> string) =>
adamc@195 348 print (^ "Leaf" pr
adamc@195 349 ::: ^ "Node" (fun _ => "")
adamc@216 350 ::: HNil) (@tree_fix A).
adamc@219 351 (** %\vspace{-.15in}% [[
adamc@219 352 = fun (A : Type) (pr : A -> string) =>
adamc@219 353 fix F (t : tree A) : string :=
adamc@219 354 match t with
adamc@219 355 | Leaf x => "Leaf" ++ "(" ++ pr x ++ ")"
adamc@219 356 | Node t1 t2 =>
adamc@219 357 "Node" ++ "(" ++ "" ++ ", " ++ F t1 ++ ", " ++ F t2 ++ ")"
adamc@219 358 end
adamc@219 359 : forall A : Type, (A -> string) -> tree A -> string
adam@302 360 ]]
adam@302 361 *)
adamc@196 362
adam@358 363 (** Some of these simplified terms seem overly complex because we have turned off simplification of calls to [append], which is what uses of the [++] operator desugar to. Selective [++] simplification would combine adjacent string literals, yielding more or less the code we would write manually to implement this printing scheme. *)
adam@358 364
adamc@196 365
adamc@196 366 (** ** Mapping *)
adamc@196 367
adamc@219 368 (** By this point, we have developed enough machinery that it is old hat to define a generic function similar to the list [map] function. *)
adamc@219 369
adamc@219 370 Definition map T dt (dd : datatypeDenote T dt) (fx : fixDenote T dt) (f : T -> T)
adamc@219 371 : T -> T :=
adamc@196 372 fx T (hmap (B1 := constructorDenote T) (B2 := constructorDenote T)
adamc@196 373 (fun _ c x r => f (c x r)) dd).
adamc@196 374
adamc@196 375 Eval compute in map Empty_set_den Empty_set_fix.
adamc@219 376 (** %\vspace{-.15in}% [[
adamc@219 377 = fun (_ : Empty_set -> Empty_set) (emp : Empty_set) =>
adamc@219 378 match emp return Empty_set with
adamc@219 379 end
adamc@219 380 : (Empty_set -> Empty_set) -> Empty_set -> Empty_set
adam@302 381 ]]
adam@302 382 *)
adamc@219 383
adamc@196 384 Eval compute in map unit_den unit_fix.
adamc@219 385 (** %\vspace{-.15in}% [[
adamc@219 386 = fun (f : unit -> unit) (_ : unit) => f tt
adamc@219 387 : (unit -> unit) -> unit -> unit
adam@302 388 ]]
adam@302 389 *)
adamc@219 390
adamc@196 391 Eval compute in map bool_den bool_fix.
adamc@219 392 (** %\vspace{-.15in}% [[
adamc@219 393 = fun (f : bool -> bool) (b : bool) => if b then f true else f false
adamc@219 394 : (bool -> bool) -> bool -> bool
adam@302 395 ]]
adam@302 396 *)
adamc@219 397
adamc@196 398 Eval compute in map nat_den nat_fix.
adamc@219 399 (** %\vspace{-.15in}% [[
adamc@219 400 = fun f : nat -> nat =>
adamc@219 401 fix F (n : nat) : nat :=
adamc@219 402 match n with
adamc@219 403 | 0%nat => f 0%nat
adamc@219 404 | S n' => f (S (F n'))
adamc@219 405 end
adamc@219 406 : (nat -> nat) -> nat -> nat
adam@302 407 ]]
adam@302 408 *)
adamc@219 409
adamc@196 410 Eval compute in fun A => map (list_den A) (@list_fix A).
adamc@219 411 (** %\vspace{-.15in}% [[
adamc@219 412 = fun (A : Type) (f : list A -> list A) =>
adamc@219 413 fix F (ls : list A) : list A :=
adamc@219 414 match ls with
adamc@219 415 | nil => f nil
adamc@219 416 | x :: ls' => f (x :: F ls')
adamc@219 417 end
adamc@219 418 : forall A : Type, (list A -> list A) -> list A -> list A
adam@302 419 ]]
adam@302 420 *)
adamc@219 421
adamc@196 422 Eval compute in fun A => map (tree_den A) (@tree_fix A).
adamc@219 423 (** %\vspace{-.15in}% [[
adamc@219 424 = fun (A : Type) (f : tree A -> tree A) =>
adamc@219 425 fix F (t : tree A) : tree A :=
adamc@219 426 match t with
adamc@219 427 | Leaf x => f (Leaf x)
adamc@219 428 | Node t1 t2 => f (Node (F t1) (F t2))
adamc@219 429 end
adamc@219 430 : forall A : Type, (tree A -> tree A) -> tree A -> tree A
adam@302 431 ]]
adam@302 432 *)
adamc@196 433
adam@358 434 (** These [map] functions are just as easy to use as those we write by hand. Can you figure out the input-output pattern that [map_nat S] displays in these examples? *)
adam@358 435
adamc@196 436 Definition map_nat := map nat_den nat_fix.
adamc@196 437 Eval simpl in map_nat S 0.
adamc@219 438 (** %\vspace{-.15in}% [[
adamc@219 439 = 1%nat
adamc@219 440 : nat
adam@302 441 ]]
adam@302 442 *)
adamc@219 443
adamc@196 444 Eval simpl in map_nat S 1.
adamc@219 445 (** %\vspace{-.15in}% [[
adamc@219 446 = 3%nat
adamc@219 447 : nat
adam@302 448 ]]
adam@302 449 *)
adamc@219 450
adamc@196 451 Eval simpl in map_nat S 2.
adamc@219 452 (** %\vspace{-.15in}% [[
adamc@219 453 = 5%nat
adamc@219 454 : nat
adam@302 455 ]]
adam@302 456 *)
adamc@196 457
adam@358 458 (** We get [map_nat S n] = [2 * n + 1], because the mapping process adds an extra [S] at every level of the inductive tree that defines a natural, including at the last level, the [O] constructor. *)
adam@358 459
adamc@196 460
adamc@196 461 (** * Proving Theorems about Recursive Definitions *)
adamc@196 462
adamc@219 463 (** We would like to be able to prove theorems about our generic functions. To do so, we need to establish additional well-formedness properties that must hold of pieces of evidence. *)
adamc@219 464
adamc@196 465 Section ok.
adamc@196 466 Variable T : Type.
adamc@196 467 Variable dt : datatype.
adamc@196 468
adamc@196 469 Variable dd : datatypeDenote T dt.
adamc@196 470 Variable fx : fixDenote T dt.
adamc@196 471
adamc@219 472 (** First, we characterize when a piece of evidence about a datatype is acceptable. The basic idea is that the type [T] should really be an inductive type with the definition given by [dd]. Semantically, inductive types are characterized by the ability to do induction on them. Therefore, we require that the usual induction principle is true, with respect to the constructors given in the encoding [dd]. *)
adamc@219 473
adamc@196 474 Definition datatypeDenoteOk :=
adamc@196 475 forall P : T -> Prop,
adamc@196 476 (forall c (m : member c dt) (x : nonrecursive c) (r : ilist T (recursive c)),
adamc@215 477 (forall i : fin (recursive c), P (get r i))
adamc@196 478 -> P ((hget dd m) x r))
adamc@196 479 -> forall v, P v.
adamc@196 480
adam@358 481 (** This definition can take a while to digest. The quantifier over [m : member c dt] is considering each constructor in turn; like in normal induction principles, each constructor has an associated proof case. The expression [hget dd m] then names the constructor we have selected. After binding [m], we quantify over all possible arguments (encoded with [x] and [r]) to the constructor that [m] selects. Within each specific case, we quantify further over [i : fin (][recursive c)] to consider all of our induction hypotheses, one for each recursive argument of the current constructor.
adamc@219 482
adamc@219 483 We have completed half the burden of defining side conditions. The other half comes in characterizing when a recursion scheme [fx] is valid. The natural condition is that [fx] behaves appropriately when applied to any constructor application. *)
adamc@219 484
adamc@196 485 Definition fixDenoteOk :=
adamc@196 486 forall (R : Type) (cases : datatypeDenote R dt)
adamc@196 487 c (m : member c dt)
adamc@196 488 (x : nonrecursive c) (r : ilist T (recursive c)),
adamc@216 489 fx cases ((hget dd m) x r)
adamc@216 490 = (hget cases m) x (imap (fx cases) r).
adamc@219 491
adamc@219 492 (** As for [datatypeDenoteOk], we consider all constructors and all possible arguments to them by quantifying over [m], [x], and [r]. The lefthand side of the equality that follows shows a call to the recursive function on the specific constructor application that we selected. The righthand side shows an application of the function case associated with constructor [m], applied to the non-recursive arguments and to appropriate recursive calls on the recursive arguments. *)
adamc@219 493
adamc@196 494 End ok.
adamc@196 495
adamc@219 496 (** We are now ready to prove that the [size] function we defined earlier always returns positive results. First, we establish a simple lemma. *)
adamc@196 497
adam@359 498 (* begin thide *)
adamc@196 499 Lemma foldr_plus : forall n (ils : ilist nat n),
adamc@196 500 foldr plus 1 ils > 0.
adamc@216 501 induction ils; crush.
adamc@196 502 Qed.
adamc@198 503 (* end thide *)
adamc@196 504
adamc@197 505 Theorem size_positive : forall T dt
adamc@197 506 (dd : datatypeDenote T dt) (fx : fixDenote T dt)
adamc@197 507 (dok : datatypeDenoteOk dd) (fok : fixDenoteOk dd fx)
adamc@196 508 (v : T),
adamc@196 509 size fx v > 0.
adamc@198 510 (* begin thide *)
adamc@219 511 unfold size; intros.
adamc@219 512 (** [[
adamc@219 513 ============================
adamc@219 514 fx nat
adamc@219 515 (hmake
adamc@219 516 (fun (x : constructor) (_ : nonrecursive x)
adamc@219 517 (r : ilist nat (recursive x)) => foldr plus 1%nat r) dt) v > 0
adamc@219 518 ]]
adamc@219 519
adamc@219 520 Our goal is an inequality over a particular call to [size], with its definition expanded. How can we proceed here? We cannot use [induction] directly, because there is no way for Coq to know that [T] is an inductive type. Instead, we need to use the induction principle encoded in our hypothesis [dok] of type [datatypeDenoteOk dd]. Let us try applying it directly.
adamc@219 521 [[
adamc@219 522 apply dok.
adam@358 523 ]]
adam@358 524 %\vspace{-.3in}%
adam@358 525 <<
adamc@219 526 Error: Impossible to unify "datatypeDenoteOk dd" with
adamc@219 527 "fx nat
adamc@219 528 (hmake
adamc@219 529 (fun (x : constructor) (_ : nonrecursive x)
adamc@219 530 (r : ilist nat (recursive x)) => foldr plus 1%nat r) dt) v > 0".
adam@358 531 >>
adamc@219 532
adam@360 533 Matching the type of [dok] with the type of our conclusion requires more than simple first-order unification, so [apply] is not up to the challenge. We can use the %\index{tactics!pattern}%[pattern] tactic to get our goal into a form that makes it apparent exactly what the induction hypothesis is. *)
adamc@219 534
adamc@219 535 pattern v.
adam@358 536 (** %\vspace{-.15in}%[[
adamc@219 537 ============================
adamc@219 538 (fun t : T =>
adamc@219 539 fx nat
adamc@219 540 (hmake
adamc@219 541 (fun (x : constructor) (_ : nonrecursive x)
adamc@219 542 (r : ilist nat (recursive x)) => foldr plus 1%nat r) dt) t > 0) v
adam@302 543 ]]
adam@302 544 *)
adamc@219 545
adamc@219 546 apply dok; crush.
adam@358 547 (** %\vspace{-.15in}%[[
adamc@219 548 H : forall i : fin (recursive c),
adamc@219 549 fx nat
adamc@219 550 (hmake
adamc@219 551 (fun (x : constructor) (_ : nonrecursive x)
adamc@219 552 (r : ilist nat (recursive x)) => foldr plus 1%nat r) dt)
adamc@219 553 (get r i) > 0
adamc@219 554 ============================
adamc@219 555 hget
adamc@219 556 (hmake
adamc@219 557 (fun (x0 : constructor) (_ : nonrecursive x0)
adamc@219 558 (r0 : ilist nat (recursive x0)) => foldr plus 1%nat r0) dt) m x
adamc@219 559 (imap
adamc@219 560 (fx nat
adamc@219 561 (hmake
adamc@219 562 (fun (x0 : constructor) (_ : nonrecursive x0)
adamc@219 563 (r0 : ilist nat (recursive x0)) =>
adamc@219 564 foldr plus 1%nat r0) dt)) r) > 0
adamc@219 565 ]]
adamc@219 566
adamc@219 567 An induction hypothesis [H] is generated, but we turn out not to need it for this example. We can simplify the goal using a library theorem about the composition of [hget] and [hmake]. *)
adamc@219 568
adamc@219 569 rewrite hget_hmake.
adam@358 570 (** %\vspace{-.15in}%[[
adamc@219 571 ============================
adamc@219 572 foldr plus 1%nat
adamc@219 573 (imap
adamc@219 574 (fx nat
adamc@219 575 (hmake
adamc@219 576 (fun (x0 : constructor) (_ : nonrecursive x0)
adamc@219 577 (r0 : ilist nat (recursive x0)) =>
adamc@219 578 foldr plus 1%nat r0) dt)) r) > 0
adamc@219 579 ]]
adamc@219 580
adamc@219 581 The lemma we proved earlier finishes the proof. *)
adamc@219 582
adamc@219 583 apply foldr_plus.
adamc@219 584
adamc@219 585 (** Using hints, we can redo this proof in a nice automated form. *)
adamc@219 586
adamc@219 587 Restart.
adamc@219 588
adam@375 589 Hint Rewrite hget_hmake.
adamc@196 590 Hint Resolve foldr_plus.
adamc@196 591
adamc@197 592 unfold size; intros; pattern v; apply dok; crush.
adamc@196 593 Qed.
adamc@198 594 (* end thide *)
adamc@197 595
adamc@219 596 (** It turned out that, in this example, we only needed to use induction degenerately as case analysis. A more involved theorem may only be proved using induction hypotheses. We will give its proof only in unautomated form and leave effective automation as an exercise for the motivated reader.
adamc@219 597
adamc@219 598 In particular, it ought to be the case that generic [map] applied to an identity function is itself an identity function. *)
adamc@219 599
adamc@197 600 Theorem map_id : forall T dt
adamc@197 601 (dd : datatypeDenote T dt) (fx : fixDenote T dt)
adamc@197 602 (dok : datatypeDenoteOk dd) (fok : fixDenoteOk dd fx)
adamc@197 603 (v : T),
adamc@197 604 map dd fx (fun x => x) v = v.
adamc@198 605 (* begin thide *)
adamc@219 606 (** Let us begin as we did in the last theorem, after adding another useful library equality as a hint. *)
adamc@219 607
adam@375 608 Hint Rewrite hget_hmap.
adamc@197 609
adamc@197 610 unfold map; intros; pattern v; apply dok; crush.
adam@358 611 (** %\vspace{-.15in}%[[
adamc@219 612 H : forall i : fin (recursive c),
adamc@219 613 fx T
adamc@219 614 (hmap
adamc@219 615 (fun (x : constructor) (c : constructorDenote T x)
adamc@219 616 (x0 : nonrecursive x) (r : ilist T (recursive x)) =>
adamc@219 617 c x0 r) dd) (get r i) = get r i
adamc@219 618 ============================
adamc@219 619 hget dd m x
adamc@219 620 (imap
adamc@219 621 (fx T
adamc@219 622 (hmap
adamc@219 623 (fun (x0 : constructor) (c0 : constructorDenote T x0)
adamc@219 624 (x1 : nonrecursive x0) (r0 : ilist T (recursive x0)) =>
adamc@219 625 c0 x1 r0) dd)) r) = hget dd m x r
adamc@219 626 ]]
adamc@197 627
adamc@219 628 Our goal is an equality whose two sides begin with the same function call and initial arguments. We believe that the remaining arguments are in fact equal as well, and the [f_equal] tactic applies this reasoning step for us formally. *)
adamc@219 629
adamc@197 630 f_equal.
adam@358 631 (** %\vspace{-.15in}%[[
adamc@219 632 ============================
adamc@219 633 imap
adamc@219 634 (fx T
adamc@219 635 (hmap
adamc@219 636 (fun (x0 : constructor) (c0 : constructorDenote T x0)
adamc@219 637 (x1 : nonrecursive x0) (r0 : ilist T (recursive x0)) =>
adamc@219 638 c0 x1 r0) dd)) r = r
adamc@219 639 ]]
adamc@219 640
adamc@219 641 At this point, it is helpful to proceed by an inner induction on the heterogeneous list [r] of recursive call results. We could arrive at a cleaner proof by breaking this step out into an explicit lemma, but here we will do the induction inline to save space.*)
adamc@219 642
adamc@219 643 induction r; crush.
adamc@219 644
adamc@219 645 (** The base case is discharged automatically, and the inductive case looks like this, where [H] is the outer IH (for induction over [T] values) and [IHn] is the inner IH (for induction over the recursive arguments).
adamc@219 646 [[
adamc@219 647 H : forall i : fin (S n),
adamc@219 648 fx T
adamc@219 649 (hmap
adamc@219 650 (fun (x : constructor) (c : constructorDenote T x)
adamc@219 651 (x0 : nonrecursive x) (r : ilist T (recursive x)) =>
adamc@219 652 c x0 r) dd)
adamc@219 653 (match i in (fin n') return ((fin (pred n') -> T) -> T) with
adamc@219 654 | First n => fun _ : fin n -> T => a
adamc@219 655 | Next n idx' => fun get_ls' : fin n -> T => get_ls' idx'
adamc@219 656 end (get r)) =
adamc@219 657 match i in (fin n') return ((fin (pred n') -> T) -> T) with
adamc@219 658 | First n => fun _ : fin n -> T => a
adamc@219 659 | Next n idx' => fun get_ls' : fin n -> T => get_ls' idx'
adamc@219 660 end (get r)
adamc@219 661 IHr : (forall i : fin n,
adamc@219 662 fx T
adamc@219 663 (hmap
adamc@219 664 (fun (x : constructor) (c : constructorDenote T x)
adamc@219 665 (x0 : nonrecursive x) (r : ilist T (recursive x)) =>
adamc@219 666 c x0 r) dd) (get r i) = get r i) ->
adamc@219 667 imap
adamc@219 668 (fx T
adamc@219 669 (hmap
adamc@219 670 (fun (x : constructor) (c : constructorDenote T x)
adamc@219 671 (x0 : nonrecursive x) (r : ilist T (recursive x)) =>
adamc@219 672 c x0 r) dd)) r = r
adamc@219 673 ============================
adamc@219 674 ICons
adamc@219 675 (fx T
adamc@219 676 (hmap
adamc@219 677 (fun (x0 : constructor) (c0 : constructorDenote T x0)
adamc@219 678 (x1 : nonrecursive x0) (r0 : ilist T (recursive x0)) =>
adamc@219 679 c0 x1 r0) dd) a)
adamc@219 680 (imap
adamc@219 681 (fx T
adamc@219 682 (hmap
adamc@219 683 (fun (x0 : constructor) (c0 : constructorDenote T x0)
adamc@219 684 (x1 : nonrecursive x0) (r0 : ilist T (recursive x0)) =>
adamc@219 685 c0 x1 r0) dd)) r) = ICons a r
adamc@219 686 ]]
adamc@219 687
adamc@219 688 We see another opportunity to apply [f_equal], this time to split our goal into two different equalities over corresponding arguments. After that, the form of the first goal matches our outer induction hypothesis [H], when we give type inference some help by specifying the right quantifier instantiation. *)
adamc@219 689
adamc@219 690 f_equal.
adamc@219 691 apply (H First).
adam@358 692 (** %\vspace{-.15in}%[[
adamc@219 693 ============================
adamc@219 694 imap
adamc@219 695 (fx T
adamc@219 696 (hmap
adamc@219 697 (fun (x0 : constructor) (c0 : constructorDenote T x0)
adamc@219 698 (x1 : nonrecursive x0) (r0 : ilist T (recursive x0)) =>
adamc@219 699 c0 x1 r0) dd)) r = r
adamc@219 700 ]]
adamc@219 701
adamc@219 702 Now the goal matches the inner IH [IHr]. *)
adamc@219 703
adamc@219 704 apply IHr; crush.
adam@358 705 (** %\vspace{-.15in}%[[
adamc@219 706 i : fin n
adamc@219 707 ============================
adamc@219 708 fx T
adamc@219 709 (hmap
adamc@219 710 (fun (x0 : constructor) (c0 : constructorDenote T x0)
adamc@219 711 (x1 : nonrecursive x0) (r0 : ilist T (recursive x0)) =>
adamc@219 712 c0 x1 r0) dd) (get r i) = get r i
adamc@219 713 ]]
adamc@219 714
adamc@219 715 We can finish the proof by applying the outer IH again, specialized to a different [fin] value. *)
adamc@219 716
adamc@216 717 apply (H (Next i)).
adamc@197 718 Qed.
adamc@198 719 (* end thide *)
adam@358 720
adam@358 721 (** The proof involves complex subgoals, but, still, few steps are required, and then we may reuse our work across a variety of datatypes. *)