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Finish Coinductive chapter
author Adam Chlipala <adamc@hcoop.net>
date Wed, 01 Oct 2008 09:56:32 -0400
parents cb135b19adb8
children a21447f76aad
rev   line source
adamc@26 1 (* Copyright (c) 2008, Adam Chlipala
adamc@26 2 *
adamc@26 3 * This work is licensed under a
adamc@26 4 * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
adamc@26 5 * Unported License.
adamc@26 6 * The license text is available at:
adamc@26 7 * http://creativecommons.org/licenses/by-nc-nd/3.0/
adamc@26 8 *)
adamc@26 9
adamc@26 10 (* begin hide *)
adamc@26 11 Require Import List.
adamc@26 12
adamc@26 13 Require Import Tactics.
adamc@26 14
adamc@26 15 Set Implicit Arguments.
adamc@26 16 (* end hide *)
adamc@26 17
adamc@26 18
adamc@43 19 (** %\chapter{Introducing Inductive Types}% *)
adamc@26 20
adamc@45 21 (** In a sense, CIC is built from just two relatively straightforward features: function types and inductive types. From this modest foundation, we can prove effectively all of the theorems of math and carry out effectively all program verifications, with enough effort expended. This chapter introduces induction and recursion for functional programming in Coq. *)
adamc@26 22
adamc@26 23
adamc@26 24 (** * Enumerations *)
adamc@26 25
adamc@26 26 (** Coq inductive types generalize the algebraic datatypes found in Haskell and ML. Confusingly enough, inductive types also generalize generalized algebraic datatypes (GADTs), by adding the possibility for type dependency. Even so, it is worth backing up from the examples of the last chapter and going over basic, algebraic datatype uses of inductive datatypes, because the chance to prove things about the values of these types adds new wrinkles beyond usual practice in Haskell and ML.
adamc@26 27
adamc@26 28 The singleton type [unit] is an inductive type: *)
adamc@26 29
adamc@26 30 Inductive unit : Set :=
adamc@26 31 | tt.
adamc@26 32
adamc@26 33 (** This vernacular command defines a new inductive type [unit] whose only value is [tt], as we can see by checking the types of the two identifiers: *)
adamc@26 34
adamc@26 35 Check unit.
adamc@26 36 (** [[
adamc@26 37
adamc@26 38 unit : Set
adamc@26 39 ]] *)
adamc@26 40 Check tt.
adamc@26 41 (** [[
adamc@26 42
adamc@26 43 tt : unit
adamc@26 44 ]] *)
adamc@26 45
adamc@26 46 (** We can prove that [unit] is a genuine singleton type. *)
adamc@26 47
adamc@26 48 Theorem unit_singleton : forall x : unit, x = tt.
adamc@26 49 (** The important thing about an inductive type is, unsurprisingly, that you can do induction over its values, and induction is the key to proving this theorem. We ask to proceed by induction on the variable [x]. *)
adamc@41 50 (* begin thide *)
adamc@26 51 induction x.
adamc@26 52 (** The goal changes to: [[
adamc@26 53
adamc@26 54 tt = tt
adamc@26 55 ]] *)
adamc@26 56 (** ...which we can discharge trivially. *)
adamc@26 57 reflexivity.
adamc@26 58 Qed.
adamc@41 59 (* end thide *)
adamc@26 60
adamc@26 61 (** It seems kind of odd to write a proof by induction with no inductive hypotheses. We could have arrived at the same result by beginning the proof with: [[
adamc@26 62
adamc@26 63 destruct x.
adamc@26 64 ...which corresponds to "proof by case analysis" in classical math. For non-recursive inductive types, the two tactics will always have identical behavior. Often case analysis is sufficient, even in proofs about recursive types, and it is nice to avoid introducing unneeded induction hypotheses.
adamc@26 65
adamc@26 66 What exactly %\textit{%#<i>#is#</i>#%}% the induction principle for [unit]? We can ask Coq: *)
adamc@26 67
adamc@26 68 Check unit_ind.
adamc@26 69 (** [[
adamc@26 70
adamc@26 71 unit_ind : forall P : unit -> Prop, P tt -> forall u : unit, P u
adamc@26 72 ]]
adamc@26 73
adamc@26 74 Every [Inductive] command defining a type [T] also defines an induction principle named [T_ind]. Coq follows the Curry-Howard correspondence and includes the ingredients of programming and proving in the same single syntactic class. Thus, our type, operations over it, and principles for reasoning about it all live in the same language and are described by the same type system. The key to telling what is a program and what is a proof lies in the distinction between the type [Prop], which appears in our induction principle; and the type [Set], which we have seen a few times already.
adamc@26 75
adamc@26 76 The convention goes like this: [Set] is the type of normal types, and the values of such types are programs. [Prop] is the type of logical propositions, and the values of such types are proofs. Thus, an induction principle has a type that shows us that it is a function for building proofs.
adamc@26 77
adamc@26 78 Specifically, [unit_ind] quantifies over a predicate [P] over [unit] values. If we can present a proof that [P] holds of [tt], then we are rewarded with a proof that [P] holds for any value [u] of type [unit]. In our last proof, the predicate was [(fun u : unit => u = tt)].
adamc@26 79
adamc@26 80 %\medskip%
adamc@26 81
adamc@26 82 We can define an inductive type even simpler than [unit]: *)
adamc@26 83
adamc@26 84 Inductive Empty_set : Set := .
adamc@26 85
adamc@26 86 (** [Empty_set] has no elements. We can prove fun theorems about it: *)
adamc@26 87
adamc@26 88 Theorem the_sky_is_falling : forall x : Empty_set, 2 + 2 = 5.
adamc@41 89 (* begin thide *)
adamc@26 90 destruct 1.
adamc@26 91 Qed.
adamc@41 92 (* end thide *)
adamc@26 93
adamc@32 94 (** Because [Empty_set] has no elements, the fact of having an element of this type implies anything. We use [destruct 1] instead of [destruct x] in the proof because unused quantified variables are relegated to being referred to by number. (There is a good reason for this, related to the unity of quantifiers and implication. An implication is just a quantification over a proof, where the quantified variable is never used. It generally makes more sense to refer to implication hypotheses by number than by name, and Coq treats our quantifier over an unused variable as an implication in determining the proper behavior.)
adamc@26 95
adamc@26 96 We can see the induction principle that made this proof so easy: *)
adamc@26 97
adamc@26 98 Check Empty_set_ind.
adamc@26 99 (** [[
adamc@26 100
adamc@26 101 Empty_set_ind : forall (P : Empty_set -> Prop) (e : Empty_set), P e
adamc@26 102 ]]
adamc@26 103
adamc@26 104 In other words, any predicate over values from the empty set holds vacuously of every such element. In the last proof, we chose the predicate [(fun _ : Empty_set => 2 + 2 = 5)].
adamc@26 105
adamc@26 106 We can also apply this get-out-of-jail-free card programmatically. Here is a lazy way of converting values of [Empty_set] to values of [unit]: *)
adamc@26 107
adamc@26 108 Definition e2u (e : Empty_set) : unit := match e with end.
adamc@26 109
adamc@26 110 (** We employ [match] pattern matching as in the last chapter. Since we match on a value whose type has no constructors, there is no need to provide any branches.
adamc@26 111
adamc@26 112 %\medskip%
adamc@26 113
adamc@26 114 Moving up the ladder of complexity, we can define the booleans: *)
adamc@26 115
adamc@26 116 Inductive bool : Set :=
adamc@26 117 | true
adamc@26 118 | false.
adamc@26 119
adamc@26 120 (** We can use less vacuous pattern matching to define boolean negation. *)
adamc@26 121
adamc@26 122 Definition not (b : bool) : bool :=
adamc@26 123 match b with
adamc@26 124 | true => false
adamc@26 125 | false => true
adamc@26 126 end.
adamc@26 127
adamc@27 128 (** An alternative definition desugars to the above: *)
adamc@27 129
adamc@27 130 Definition not' (b : bool) : bool :=
adamc@27 131 if b then false else true.
adamc@27 132
adamc@26 133 (** We might want to prove that [not] is its own inverse operation. *)
adamc@26 134
adamc@26 135 Theorem not_inverse : forall b : bool, not (not b) = b.
adamc@41 136 (* begin thide *)
adamc@26 137 destruct b.
adamc@26 138
adamc@26 139 (** After we case analyze on [b], we are left with one subgoal for each constructor of [bool].
adamc@26 140
adamc@26 141 [[
adamc@26 142
adamc@26 143 2 subgoals
adamc@26 144
adamc@26 145 ============================
adamc@26 146 not (not true) = true
adamc@26 147 ]]
adamc@26 148
adamc@26 149 [[
adamc@26 150 subgoal 2 is:
adamc@26 151 not (not false) = false
adamc@26 152 ]]
adamc@26 153
adamc@26 154 The first subgoal follows by Coq's rules of computation, so we can dispatch it easily: *)
adamc@26 155
adamc@26 156 reflexivity.
adamc@26 157
adamc@26 158 (** Likewise for the second subgoal, so we can restart the proof and give a very compact justification. *)
adamc@26 159
adamc@26 160 Restart.
adamc@26 161 destruct b; reflexivity.
adamc@26 162 Qed.
adamc@41 163 (* end thide *)
adamc@27 164
adamc@27 165 (** Another theorem about booleans illustrates another useful tactic. *)
adamc@27 166
adamc@27 167 Theorem not_ineq : forall b : bool, not b <> b.
adamc@41 168 (* begin thide *)
adamc@27 169 destruct b; discriminate.
adamc@27 170 Qed.
adamc@41 171 (* end thide *)
adamc@27 172
adamc@27 173 (** [discriminate] is used to prove that two values of an inductive type are not equal, whenever the values are formed with different constructors. In this case, the different constructors are [true] and [false].
adamc@27 174
adamc@27 175 At this point, it is probably not hard to guess what the underlying induction principle for [bool] is. *)
adamc@27 176
adamc@27 177 Check bool_ind.
adamc@27 178 (** [[
adamc@27 179
adamc@27 180 bool_ind : forall P : bool -> Prop, P true -> P false -> forall b : bool, P b
adamc@27 181 ]] *)
adamc@28 182
adamc@28 183
adamc@28 184 (** * Simple Recursive Types *)
adamc@28 185
adamc@28 186 (** The natural numbers are the simplest common example of an inductive type that actually deserves the name. *)
adamc@28 187
adamc@28 188 Inductive nat : Set :=
adamc@28 189 | O : nat
adamc@28 190 | S : nat -> nat.
adamc@28 191
adamc@28 192 (** [O] is zero, and [S] is the successor function, so that [0] is syntactic sugar for [O], [1] for [S O], [2] for [S (S O)], and so on.
adamc@28 193
adamc@28 194 Pattern matching works as we demonstrated in the last chapter: *)
adamc@28 195
adamc@28 196 Definition isZero (n : nat) : bool :=
adamc@28 197 match n with
adamc@28 198 | O => true
adamc@28 199 | S _ => false
adamc@28 200 end.
adamc@28 201
adamc@28 202 Definition pred (n : nat) : nat :=
adamc@28 203 match n with
adamc@28 204 | O => O
adamc@28 205 | S n' => n'
adamc@28 206 end.
adamc@28 207
adamc@28 208 (** We can prove theorems by case analysis: *)
adamc@28 209
adamc@28 210 Theorem S_isZero : forall n : nat, isZero (pred (S (S n))) = false.
adamc@41 211 (* begin thide *)
adamc@28 212 destruct n; reflexivity.
adamc@28 213 Qed.
adamc@41 214 (* end thide *)
adamc@28 215
adamc@28 216 (** We can also now get into genuine inductive theorems. First, we will need a recursive function, to make things interesting. *)
adamc@28 217
adamc@28 218 Fixpoint plus (n m : nat) {struct n} : nat :=
adamc@28 219 match n with
adamc@28 220 | O => m
adamc@28 221 | S n' => S (plus n' m)
adamc@28 222 end.
adamc@28 223
adamc@28 224 (** Recall that [Fixpoint] is Coq's mechanism for recursive function definitions, and that the [{struct n}] annotation is noting which function argument decreases structurally at recursive calls.
adamc@28 225
adamc@28 226 Some theorems about [plus] can be proved without induction. *)
adamc@28 227
adamc@28 228 Theorem O_plus_n : forall n : nat, plus O n = n.
adamc@41 229 (* begin thide *)
adamc@28 230 intro; reflexivity.
adamc@28 231 Qed.
adamc@41 232 (* end thide *)
adamc@28 233
adamc@28 234 (** Coq's computation rules automatically simplify the application of [plus]. If we just reverse the order of the arguments, though, this no longer works, and we need induction. *)
adamc@28 235
adamc@28 236 Theorem n_plus_O : forall n : nat, plus n O = n.
adamc@41 237 (* begin thide *)
adamc@28 238 induction n.
adamc@28 239
adamc@28 240 (** Our first subgoal is [plus O O = O], which %\textit{%#<i>#is#</i>#%}% trivial by computation. *)
adamc@28 241
adamc@28 242 reflexivity.
adamc@28 243
adamc@28 244 (** Our second subgoal is more work and also demonstrates our first inductive hypothesis.
adamc@28 245
adamc@28 246 [[
adamc@28 247
adamc@28 248 n : nat
adamc@28 249 IHn : plus n O = n
adamc@28 250 ============================
adamc@28 251 plus (S n) O = S n
adamc@28 252 ]]
adamc@28 253
adamc@28 254 We can start out by using computation to simplify the goal as far as we can. *)
adamc@28 255
adamc@28 256 simpl.
adamc@28 257
adamc@28 258 (** Now the conclusion is [S (plus n O) = S n]. Using our inductive hypothesis: *)
adamc@28 259
adamc@28 260 rewrite IHn.
adamc@28 261
adamc@28 262 (** ...we get a trivial conclusion [S n = S n]. *)
adamc@28 263
adamc@28 264 reflexivity.
adamc@28 265
adamc@28 266 (** Not much really went on in this proof, so the [crush] tactic from the [Tactics] module can prove this theorem automatically. *)
adamc@28 267
adamc@28 268 Restart.
adamc@28 269 induction n; crush.
adamc@28 270 Qed.
adamc@41 271 (* end thide *)
adamc@28 272
adamc@28 273 (** We can check out the induction principle at work here: *)
adamc@28 274
adamc@28 275 Check nat_ind.
adamc@28 276 (** [[
adamc@28 277
adamc@28 278 nat_ind : forall P : nat -> Prop,
adamc@28 279 P O -> (forall n : nat, P n -> P (S n)) -> forall n : nat, P n
adamc@28 280 ]]
adamc@28 281
adamc@28 282 Each of the two cases of our last proof came from the type of one of the arguments to [nat_ind]. We chose [P] to be [(fun n : nat => plus n O = n)]. The first proof case corresponded to [P O], and the second case to [(forall n : nat, P n -> P (S n))]. The free variable [n] and inductive hypothesis [IHn] came from the argument types given here.
adamc@28 283
adamc@28 284 Since [nat] has a constructor that takes an argument, we may sometimes need to know that that constructor is injective. *)
adamc@28 285
adamc@28 286 Theorem S_inj : forall n m : nat, S n = S m -> n = m.
adamc@41 287 (* begin thide *)
adamc@28 288 injection 1; trivial.
adamc@28 289 Qed.
adamc@41 290 (* end thide *)
adamc@28 291
adamc@28 292 (** [injection] refers to a premise by number, adding new equalities between the corresponding arguments of equated terms that are formed with the same constructor. We end up needing to prove [n = m -> n = m], so it is unsurprising that a tactic named [trivial] is able to finish the proof.
adamc@28 293
adamc@29 294 There is also a very useful tactic called [congruence] that can prove this theorem immediately. [congruence] generalizes [discriminate] and [injection], and it also adds reasoning about the general properties of equality, such as that a function returns equal results on equal arguments. That is, [congruence] is a %\textit{%#<i>#complete decision procedure for the theory of equality and uninterpreted functions#</i>#%}%, plus some smarts about inductive types.
adamc@29 295
adamc@29 296 %\medskip%
adamc@29 297
adamc@29 298 We can define a type of lists of natural numbers. *)
adamc@29 299
adamc@29 300 Inductive nat_list : Set :=
adamc@29 301 | NNil : nat_list
adamc@29 302 | NCons : nat -> nat_list -> nat_list.
adamc@29 303
adamc@29 304 (** Recursive definitions are straightforward extensions of what we have seen before. *)
adamc@29 305
adamc@29 306 Fixpoint nlength (ls : nat_list) : nat :=
adamc@29 307 match ls with
adamc@29 308 | NNil => O
adamc@29 309 | NCons _ ls' => S (nlength ls')
adamc@29 310 end.
adamc@29 311
adamc@29 312 Fixpoint napp (ls1 ls2 : nat_list) {struct ls1} : nat_list :=
adamc@29 313 match ls1 with
adamc@29 314 | NNil => ls2
adamc@29 315 | NCons n ls1' => NCons n (napp ls1' ls2)
adamc@29 316 end.
adamc@29 317
adamc@29 318 (** Inductive theorem proving can again be automated quite effectively. *)
adamc@29 319
adamc@29 320 Theorem nlength_napp : forall ls1 ls2 : nat_list, nlength (napp ls1 ls2)
adamc@29 321 = plus (nlength ls1) (nlength ls2).
adamc@41 322 (* begin thide *)
adamc@29 323 induction ls1; crush.
adamc@29 324 Qed.
adamc@41 325 (* end thide *)
adamc@29 326
adamc@29 327 Check nat_list_ind.
adamc@29 328 (** [[
adamc@29 329
adamc@29 330 nat_list_ind
adamc@29 331 : forall P : nat_list -> Prop,
adamc@29 332 P NNil ->
adamc@29 333 (forall (n : nat) (n0 : nat_list), P n0 -> P (NCons n n0)) ->
adamc@29 334 forall n : nat_list, P n
adamc@29 335 ]]
adamc@29 336
adamc@29 337 %\medskip%
adamc@29 338
adamc@29 339 In general, we can implement any "tree" types as inductive types. For example, here are binary trees of naturals. *)
adamc@29 340
adamc@29 341 Inductive nat_btree : Set :=
adamc@29 342 | NLeaf : nat_btree
adamc@29 343 | NNode : nat_btree -> nat -> nat_btree -> nat_btree.
adamc@29 344
adamc@29 345 Fixpoint nsize (tr : nat_btree) : nat :=
adamc@29 346 match tr with
adamc@35 347 | NLeaf => S O
adamc@29 348 | NNode tr1 _ tr2 => plus (nsize tr1) (nsize tr2)
adamc@29 349 end.
adamc@29 350
adamc@29 351 Fixpoint nsplice (tr1 tr2 : nat_btree) {struct tr1} : nat_btree :=
adamc@29 352 match tr1 with
adamc@35 353 | NLeaf => NNode tr2 O NLeaf
adamc@29 354 | NNode tr1' n tr2' => NNode (nsplice tr1' tr2) n tr2'
adamc@29 355 end.
adamc@29 356
adamc@29 357 Theorem plus_assoc : forall n1 n2 n3 : nat, plus (plus n1 n2) n3 = plus n1 (plus n2 n3).
adamc@41 358 (* begin thide *)
adamc@29 359 induction n1; crush.
adamc@29 360 Qed.
adamc@41 361 (* end thide *)
adamc@29 362
adamc@29 363 Theorem nsize_nsplice : forall tr1 tr2 : nat_btree, nsize (nsplice tr1 tr2)
adamc@29 364 = plus (nsize tr2) (nsize tr1).
adamc@41 365 (* begin thide *)
adamc@29 366 Hint Rewrite n_plus_O plus_assoc : cpdt.
adamc@29 367
adamc@29 368 induction tr1; crush.
adamc@29 369 Qed.
adamc@41 370 (* end thide *)
adamc@29 371
adamc@29 372 Check nat_btree_ind.
adamc@29 373 (** [[
adamc@29 374
adamc@29 375 nat_btree_ind
adamc@29 376 : forall P : nat_btree -> Prop,
adamc@29 377 P NLeaf ->
adamc@29 378 (forall n : nat_btree,
adamc@29 379 P n -> forall (n0 : nat) (n1 : nat_btree), P n1 -> P (NNode n n0 n1)) ->
adamc@29 380 forall n : nat_btree, P n
adamc@29 381 ]] *)
adamc@30 382
adamc@30 383
adamc@30 384 (** * Parameterized Types *)
adamc@30 385
adamc@30 386 (** We can also define polymorphic inductive types, as with algebraic datatypes in Haskell and ML. *)
adamc@30 387
adamc@30 388 Inductive list (T : Set) : Set :=
adamc@30 389 | Nil : list T
adamc@30 390 | Cons : T -> list T -> list T.
adamc@30 391
adamc@30 392 Fixpoint length T (ls : list T) : nat :=
adamc@30 393 match ls with
adamc@30 394 | Nil => O
adamc@30 395 | Cons _ ls' => S (length ls')
adamc@30 396 end.
adamc@30 397
adamc@30 398 Fixpoint app T (ls1 ls2 : list T) {struct ls1} : list T :=
adamc@30 399 match ls1 with
adamc@30 400 | Nil => ls2
adamc@30 401 | Cons x ls1' => Cons x (app ls1' ls2)
adamc@30 402 end.
adamc@30 403
adamc@30 404 Theorem length_app : forall T (ls1 ls2 : list T), length (app ls1 ls2)
adamc@30 405 = plus (length ls1) (length ls2).
adamc@41 406 (* begin thide *)
adamc@30 407 induction ls1; crush.
adamc@30 408 Qed.
adamc@41 409 (* end thide *)
adamc@30 410
adamc@30 411 (** There is a useful shorthand for writing many definitions that share the same parameter, based on Coq's %\textit{%#<i>#section#</i>#%}% mechanism. The following block of code is equivalent to the above: *)
adamc@30 412
adamc@30 413 (* begin hide *)
adamc@30 414 Reset list.
adamc@30 415 (* end hide *)
adamc@30 416
adamc@30 417 Section list.
adamc@30 418 Variable T : Set.
adamc@30 419
adamc@30 420 Inductive list : Set :=
adamc@30 421 | Nil : list
adamc@30 422 | Cons : T -> list -> list.
adamc@30 423
adamc@30 424 Fixpoint length (ls : list) : nat :=
adamc@30 425 match ls with
adamc@30 426 | Nil => O
adamc@30 427 | Cons _ ls' => S (length ls')
adamc@30 428 end.
adamc@30 429
adamc@30 430 Fixpoint app (ls1 ls2 : list) {struct ls1} : list :=
adamc@30 431 match ls1 with
adamc@30 432 | Nil => ls2
adamc@30 433 | Cons x ls1' => Cons x (app ls1' ls2)
adamc@30 434 end.
adamc@30 435
adamc@30 436 Theorem length_app : forall ls1 ls2 : list, length (app ls1 ls2)
adamc@30 437 = plus (length ls1) (length ls2).
adamc@41 438 (* begin thide *)
adamc@30 439 induction ls1; crush.
adamc@30 440 Qed.
adamc@41 441 (* end thide *)
adamc@30 442 End list.
adamc@30 443
adamc@35 444 (* begin hide *)
adamc@35 445 Implicit Arguments Nil [T].
adamc@35 446 (* end hide *)
adamc@35 447
adamc@30 448 (** After we end the section, the [Variable]s we used are added as extra function parameters for each defined identifier, as needed. *)
adamc@30 449
adamc@30 450 Check list.
adamc@30 451 (** [[
adamc@30 452
adamc@30 453 list
adamc@30 454 : Set -> Set
adamc@30 455 ]] *)
adamc@30 456
adamc@30 457 Check Cons.
adamc@30 458 (** [[
adamc@30 459
adamc@30 460 Cons
adamc@30 461 : forall T : Set, T -> list T -> list T
adamc@30 462 ]] *)
adamc@30 463
adamc@30 464 Check length.
adamc@30 465 (** [[
adamc@30 466
adamc@30 467 length
adamc@30 468 : forall T : Set, list T -> nat
adamc@30 469 ]]
adamc@30 470
adamc@30 471 The extra parameter [T] is treated as a new argument to the induction principle, too. *)
adamc@30 472
adamc@30 473 Check list_ind.
adamc@30 474 (** [[
adamc@30 475
adamc@30 476 list_ind
adamc@30 477 : forall (T : Set) (P : list T -> Prop),
adamc@30 478 P (Nil T) ->
adamc@30 479 (forall (t : T) (l : list T), P l -> P (Cons t l)) ->
adamc@30 480 forall l : list T, P l
adamc@30 481 ]]
adamc@30 482
adamc@30 483 Thus, even though we just saw that [T] is added as an extra argument to the constructor [Cons], there is no quantifier for [T] in the type of the inductive case like there is for each of the other arguments. *)
adamc@31 484
adamc@31 485
adamc@31 486 (** * Mutually Inductive Types *)
adamc@31 487
adamc@31 488 (** We can define inductive types that refer to each other: *)
adamc@31 489
adamc@31 490 Inductive even_list : Set :=
adamc@31 491 | ENil : even_list
adamc@31 492 | ECons : nat -> odd_list -> even_list
adamc@31 493
adamc@31 494 with odd_list : Set :=
adamc@31 495 | OCons : nat -> even_list -> odd_list.
adamc@31 496
adamc@31 497 Fixpoint elength (el : even_list) : nat :=
adamc@31 498 match el with
adamc@31 499 | ENil => O
adamc@31 500 | ECons _ ol => S (olength ol)
adamc@31 501 end
adamc@31 502
adamc@31 503 with olength (ol : odd_list) : nat :=
adamc@31 504 match ol with
adamc@31 505 | OCons _ el => S (elength el)
adamc@31 506 end.
adamc@31 507
adamc@31 508 Fixpoint eapp (el1 el2 : even_list) {struct el1} : even_list :=
adamc@31 509 match el1 with
adamc@31 510 | ENil => el2
adamc@31 511 | ECons n ol => ECons n (oapp ol el2)
adamc@31 512 end
adamc@31 513
adamc@31 514 with oapp (ol : odd_list) (el : even_list) {struct ol} : odd_list :=
adamc@31 515 match ol with
adamc@31 516 | OCons n el' => OCons n (eapp el' el)
adamc@31 517 end.
adamc@31 518
adamc@31 519 (** Everything is going roughly the same as in past examples, until we try to prove a theorem similar to those that came before. *)
adamc@31 520
adamc@31 521 Theorem elength_eapp : forall el1 el2 : even_list,
adamc@31 522 elength (eapp el1 el2) = plus (elength el1) (elength el2).
adamc@41 523 (* begin thide *)
adamc@31 524 induction el1; crush.
adamc@31 525
adamc@31 526 (** One goal remains: [[
adamc@31 527
adamc@31 528 n : nat
adamc@31 529 o : odd_list
adamc@31 530 el2 : even_list
adamc@31 531 ============================
adamc@31 532 S (olength (oapp o el2)) = S (plus (olength o) (elength el2))
adamc@31 533 ]]
adamc@31 534
adamc@31 535 We have no induction hypothesis, so we cannot prove this goal without starting another induction, which would reach a similar point, sending us into a futile infinite chain of inductions. The problem is that Coq's generation of [T_ind] principles is incomplete. We only get non-mutual induction principles generated by default. *)
adamc@31 536
adamc@31 537 Abort.
adamc@31 538 Check even_list_ind.
adamc@31 539 (** [[
adamc@31 540
adamc@31 541 even_list_ind
adamc@31 542 : forall P : even_list -> Prop,
adamc@31 543 P ENil ->
adamc@31 544 (forall (n : nat) (o : odd_list), P (ECons n o)) ->
adamc@31 545 forall e : even_list, P e
adamc@31 546 ]]
adamc@31 547
adamc@31 548 We see that no inductive hypotheses are included anywhere in the type. To get them, we must ask for mutual principles as we need them, using the [Scheme] command. *)
adamc@31 549
adamc@31 550 Scheme even_list_mut := Induction for even_list Sort Prop
adamc@31 551 with odd_list_mut := Induction for odd_list Sort Prop.
adamc@31 552
adamc@31 553 Check even_list_mut.
adamc@31 554 (** [[
adamc@31 555
adamc@31 556 even_list_mut
adamc@31 557 : forall (P : even_list -> Prop) (P0 : odd_list -> Prop),
adamc@31 558 P ENil ->
adamc@31 559 (forall (n : nat) (o : odd_list), P0 o -> P (ECons n o)) ->
adamc@31 560 (forall (n : nat) (e : even_list), P e -> P0 (OCons n e)) ->
adamc@31 561 forall e : even_list, P e
adamc@31 562 ]]
adamc@31 563
adamc@31 564 This is the principle we wanted in the first place. There is one more wrinkle left in using it: the [induction] tactic will not apply it for us automatically. It will be helpful to look at how to prove one of our past examples without using [induction], so that we can then generalize the technique to mutual inductive types. *)
adamc@31 565
adamc@31 566 Theorem n_plus_O' : forall n : nat, plus n O = n.
adamc@31 567 apply (nat_ind (fun n => plus n O = n)); crush.
adamc@31 568 Qed.
adamc@31 569
adamc@31 570 (** From this example, we can see that [induction] is not magic. It only does some bookkeeping for us to make it easy to apply a theorem, which we can do directly with the [apply] tactic. We apply not just an identifier but a partial application of it, specifying the predicate we mean to prove holds for all naturals.
adamc@31 571
adamc@31 572 This technique generalizes to our mutual example: *)
adamc@31 573
adamc@31 574 Theorem elength_eapp : forall el1 el2 : even_list,
adamc@31 575 elength (eapp el1 el2) = plus (elength el1) (elength el2).
adamc@41 576
adamc@31 577 apply (even_list_mut
adamc@31 578 (fun el1 : even_list => forall el2 : even_list,
adamc@31 579 elength (eapp el1 el2) = plus (elength el1) (elength el2))
adamc@31 580 (fun ol : odd_list => forall el : even_list,
adamc@31 581 olength (oapp ol el) = plus (olength ol) (elength el))); crush.
adamc@31 582 Qed.
adamc@41 583 (* end thide *)
adamc@31 584
adamc@31 585 (** We simply need to specify two predicates, one for each of the mutually inductive types. In general, it would not be a good idea to assume that a proof assistant could infer extra predicates, so this way of applying mutual induction is about as straightforward as we could hope for. *)
adamc@33 586
adamc@33 587
adamc@33 588 (** * Reflexive Types *)
adamc@33 589
adamc@33 590 (** A kind of inductive type called a %\textit{%#<i>#reflexive type#</i>#%}% is defined in terms of functions that have the type being defined as their range. One very useful class of examples is in modeling variable binders. For instance, here is a type for encoding the syntax of a subset of first-order logic: *)
adamc@33 591
adamc@33 592 Inductive formula : Set :=
adamc@33 593 | Eq : nat -> nat -> formula
adamc@33 594 | And : formula -> formula -> formula
adamc@33 595 | Forall : (nat -> formula) -> formula.
adamc@33 596
adamc@33 597 (** Our kinds of formulas are equalities between naturals, conjunction, and universal quantification over natural numbers. We avoid needing to include a notion of "variables" in our type, by using Coq functions to encode quantification. For instance, here is the encoding of [forall x : nat, x = x]: *)
adamc@33 598
adamc@33 599 Example forall_refl : formula := Forall (fun x => Eq x x).
adamc@33 600
adamc@33 601 (** We can write recursive functions over reflexive types quite naturally. Here is one translating our formulas into native Coq propositions. *)
adamc@33 602
adamc@33 603 Fixpoint formulaDenote (f : formula) : Prop :=
adamc@33 604 match f with
adamc@33 605 | Eq n1 n2 => n1 = n2
adamc@33 606 | And f1 f2 => formulaDenote f1 /\ formulaDenote f2
adamc@33 607 | Forall f' => forall n : nat, formulaDenote (f' n)
adamc@33 608 end.
adamc@33 609
adamc@33 610 (** We can also encode a trivial formula transformation that swaps the order of equality and conjunction operands. *)
adamc@33 611
adamc@33 612 Fixpoint swapper (f : formula) : formula :=
adamc@33 613 match f with
adamc@33 614 | Eq n1 n2 => Eq n2 n1
adamc@33 615 | And f1 f2 => And (swapper f2) (swapper f1)
adamc@33 616 | Forall f' => Forall (fun n => swapper (f' n))
adamc@33 617 end.
adamc@33 618
adamc@33 619 (** It is helpful to prove that this transformation does not make true formulas false. *)
adamc@33 620
adamc@33 621 Theorem swapper_preserves_truth : forall f, formulaDenote f -> formulaDenote (swapper f).
adamc@41 622 (* begin thide *)
adamc@33 623 induction f; crush.
adamc@33 624 Qed.
adamc@41 625 (* end thide *)
adamc@33 626
adamc@33 627 (** We can take a look at the induction principle behind this proof. *)
adamc@33 628
adamc@33 629 Check formula_ind.
adamc@33 630 (** [[
adamc@33 631
adamc@33 632 formula_ind
adamc@33 633 : forall P : formula -> Prop,
adamc@33 634 (forall n n0 : nat, P (Eq n n0)) ->
adamc@33 635 (forall f0 : formula,
adamc@33 636 P f0 -> forall f1 : formula, P f1 -> P (And f0 f1)) ->
adamc@33 637 (forall f1 : nat -> formula,
adamc@33 638 (forall n : nat, P (f1 n)) -> P (Forall f1)) ->
adamc@33 639 forall f2 : formula, P f2
adamc@33 640 ]] *)
adamc@33 641
adamc@33 642 (** Focusing on the [Forall] case, which comes third, we see that we are allowed to assume that the theorem holds %\textit{%#<i>#for any application of the argument function [f1]#</i>#%}%. That is, Coq induction principles do not follow a simple rule that the textual representations of induction variables must get shorter in appeals to induction hypotheses. Luckily for us, the people behind the metatheory of Coq have verified that this flexibility does not introduce unsoundness.
adamc@33 643
adamc@33 644 %\medskip%
adamc@33 645
adamc@33 646 Up to this point, we have seen how to encode in Coq more and more of what is possible with algebraic datatypes in Haskell and ML. This may have given the inaccurate impression that inductive types are a strict extension of algebraic datatypes. In fact, Coq must rule out some types allowed by Haskell and ML, for reasons of soundness. Reflexive types provide our first good example of such a case.
adamc@33 647
adamc@33 648 Given our last example of an inductive type, many readers are probably eager to try encoding the syntax of lambda calculus. Indeed, the function-based representation technique that we just used, called %\textit{%#<i>#higher-order abstract syntax (HOAS)#</i>#%}%, is the representation of choice for lambda calculi in Twelf and in many applications implemented in Haskell and ML. Let us try to import that choice to Coq: *)
adamc@33 649
adamc@33 650 (** [[
adamc@33 651
adamc@33 652 Inductive term : Set :=
adamc@33 653 | App : term -> term -> term
adamc@33 654 | Abs : (term -> term) -> term.
adamc@33 655
adamc@33 656 [[
adamc@33 657 Error: Non strictly positive occurrence of "term" in "(term -> term) -> term"
adamc@33 658 ]]
adamc@33 659
adamc@33 660 We have run afoul of the %\textit{%#<i>#strict positivity requirement#</i>#%}% for inductive definitions, which says that the type being defined may not occur to the left of an arrow in the type of a constructor argument. It is important that the type of a constructor is viewed in terms of a series of arguments and a result, since obviously we need recursive occurrences to the lefts of the outermost arrows if we are to have recursive occurrences at all.
adamc@33 661
adamc@33 662 Why must Coq enforce this restriction? Imagine that our last definition had been accepted, allowing us to write this function:
adamc@33 663
adamc@33 664 [[
adamc@33 665 Definition uhoh (t : term) : term :=
adamc@33 666 match t with
adamc@33 667 | Abs f => f t
adamc@33 668 | _ => t
adamc@33 669 end.
adamc@33 670
adamc@33 671 Using an informal idea of Coq's semantics, it is easy to verify that the application [uhoh (Abs uhoh)] will run forever. This would be a mere curiosity in OCaml and Haskell, where non-termination is commonplace, though the fact that we have a non-terminating program without explicit recursive function definitions is unusual.
adamc@33 672
adamc@33 673 For Coq, however, this would be a disaster. The possibility of writing such a function would destroy all our confidence that proving a theorem means anything. Since Coq combines programs and proofs in one language, we would be able to prove every theorem with an infinite loop.
adamc@33 674
adamc@33 675 Nonetheless, the basic insight of HOAS is a very useful one, and there are ways to realize most benefits of HOAS in Coq. We will study a particular technique of this kind in the later chapters on programming language syntax and semantics. *)
adamc@34 676
adamc@34 677
adamc@34 678 (** * An Interlude on Proof Terms *)
adamc@34 679
adamc@34 680 (** As we have emphasized a few times already, Coq proofs are actually programs, written in the same language we have been using in our examples all along. We can get a first sense of what this means by taking a look at the definitions of some of the induction principles we have used. *)
adamc@34 681
adamc@34 682 Print unit_ind.
adamc@34 683 (** [[
adamc@34 684
adamc@34 685 unit_ind =
adamc@34 686 fun P : unit -> Prop => unit_rect P
adamc@34 687 : forall P : unit -> Prop, P tt -> forall u : unit, P u
adamc@34 688 ]]
adamc@34 689
adamc@34 690 We see that this induction principle is defined in terms of a more general principle, [unit_rect]. *)
adamc@34 691
adamc@34 692 Check unit_rect.
adamc@34 693 (** [[
adamc@34 694
adamc@34 695 unit_rect
adamc@34 696 : forall P : unit -> Type, P tt -> forall u : unit, P u
adamc@34 697 ]]
adamc@34 698
adamc@34 699 [unit_rect] gives [P] type [unit -> Type] instead of [unit -> Prop]. [Type] is another universe, like [Set] and [Prop]. In fact, it is a common supertype of both. Later on, we will discuss exactly what the significances of the different universes are. For now, it is just important that we can use [Type] as a sort of meta-universe that may turn out to be either [Set] or [Prop]. We can see the symmetry inherent in the subtyping relationship by printing the definition of another principle that was generated for [unit] automatically: *)
adamc@34 700
adamc@34 701 Print unit_rec.
adamc@34 702 (** [[
adamc@34 703
adamc@34 704 unit_rec =
adamc@34 705 fun P : unit -> Set => unit_rect P
adamc@34 706 : forall P : unit -> Set, P tt -> forall u : unit, P u
adamc@34 707 ]]
adamc@34 708
adamc@34 709 This is identical to the definition for [unit_ind], except that we have substituted [Set] for [Prop]. For most inductive types [T], then, we get not just induction principles [T_ind], but also recursion principles [T_rec]. We can use [T_rec] to write recursive definitions without explicit [Fixpoint] recursion. For instance, the following two definitions are equivalent: *)
adamc@34 710
adamc@34 711 Definition always_O (u : unit) : nat :=
adamc@34 712 match u with
adamc@34 713 | tt => O
adamc@34 714 end.
adamc@34 715
adamc@34 716 Definition always_O' (u : unit) : nat :=
adamc@34 717 unit_rec (fun _ : unit => nat) O u.
adamc@34 718
adamc@34 719 (** Going even further down the rabbit hole, [unit_rect] itself is not even a primitive. It is a functional program that we can write manually. *)
adamc@34 720
adamc@34 721 Print unit_rect.
adamc@34 722
adamc@34 723 (** [[
adamc@34 724
adamc@34 725 unit_rect =
adamc@34 726 fun (P : unit -> Type) (f : P tt) (u : unit) =>
adamc@34 727 match u as u0 return (P u0) with
adamc@34 728 | tt => f
adamc@34 729 end
adamc@34 730 : forall P : unit -> Type, P tt -> forall u : unit, P u
adamc@34 731 ]]
adamc@34 732
adamc@34 733 The only new feature we see is an [as] clause for a [match], which is used in concert with the [return] clause that we saw in the introduction. Since the type of the [match] is dependent on the value of the object being analyzed, we must give that object a name so that we can refer to it in the [return] clause.
adamc@34 734
adamc@34 735 To prove that [unit_rect] is nothing special, we can reimplement it manually. *)
adamc@34 736
adamc@34 737 Definition unit_rect' (P : unit -> Type) (f : P tt) (u : unit) :=
adamc@34 738 match u return (P u) with
adamc@34 739 | tt => f
adamc@34 740 end.
adamc@34 741
adamc@34 742 (** We use the handy shorthand that lets us omit an [as] annotation when matching on a variable, simply using that variable directly in the [return] clause.
adamc@34 743
adamc@34 744 We can check the implement of [nat_rect] as well: *)
adamc@34 745
adamc@34 746 Print nat_rect.
adamc@34 747 (** [[
adamc@34 748
adamc@34 749 nat_rect =
adamc@34 750 fun (P : nat -> Type) (f : P O) (f0 : forall n : nat, P n -> P (S n)) =>
adamc@34 751 fix F (n : nat) : P n :=
adamc@34 752 match n as n0 return (P n0) with
adamc@34 753 | O => f
adamc@34 754 | S n0 => f0 n0 (F n0)
adamc@34 755 end
adamc@34 756 : forall P : nat -> Type,
adamc@34 757 P O -> (forall n : nat, P n -> P (S n)) -> forall n : nat, P n
adamc@34 758 ]]
adamc@34 759
adamc@34 760 Now we have an actual recursive definition. [fix] expressions are an anonymous form of [Fixpoint], just as [fun] expressions stand for anonymous non-recursive functions. Beyond that, the syntax of [fix] mirrors that of [Fixpoint]. We can understand the definition of [nat_rect] better by reimplementing [nat_ind] using sections. *)
adamc@34 761
adamc@34 762 Section nat_ind'.
adamc@34 763 (** First, we have the property of natural numbers that we aim to prove. *)
adamc@34 764 Variable P : nat -> Prop.
adamc@34 765
adamc@34 766 (** Then we require a proof of the [O] case. *)
adamc@38 767 Hypothesis O_case : P O.
adamc@34 768
adamc@34 769 (** Next is a proof of the [S] case, which may assume an inductive hypothesis. *)
adamc@38 770 Hypothesis S_case : forall n : nat, P n -> P (S n).
adamc@34 771
adamc@34 772 (** Finally, we define a recursive function to tie the pieces together. *)
adamc@34 773 Fixpoint nat_ind' (n : nat) : P n :=
adamc@34 774 match n return (P n) with
adamc@34 775 | O => O_case
adamc@34 776 | S n' => S_case (nat_ind' n')
adamc@34 777 end.
adamc@34 778 End nat_ind'.
adamc@34 779
adamc@38 780 (** Closing the section adds the [Variable]s and [Hypothesis]es as new [fun]-bound arguments to [nat_ind'], and, modulo the use of [Prop] instead of [Type], we end up with the exact same definition that was generated automatically for [nat_rect].
adamc@34 781
adamc@34 782 %\medskip%
adamc@34 783
adamc@34 784 We can also examine the definition of [even_list_mut], which we generated with [Scheme] for a mutually-recursive type. *)
adamc@34 785
adamc@34 786 Print even_list_mut.
adamc@34 787 (** [[
adamc@34 788
adamc@34 789 even_list_mut =
adamc@34 790 fun (P : even_list -> Prop) (P0 : odd_list -> Prop)
adamc@34 791 (f : P ENil) (f0 : forall (n : nat) (o : odd_list), P0 o -> P (ECons n o))
adamc@34 792 (f1 : forall (n : nat) (e : even_list), P e -> P0 (OCons n e)) =>
adamc@34 793 fix F (e : even_list) : P e :=
adamc@34 794 match e as e0 return (P e0) with
adamc@34 795 | ENil => f
adamc@34 796 | ECons n o => f0 n o (F0 o)
adamc@34 797 end
adamc@34 798 with F0 (o : odd_list) : P0 o :=
adamc@34 799 match o as o0 return (P0 o0) with
adamc@34 800 | OCons n e => f1 n e (F e)
adamc@34 801 end
adamc@34 802 for F
adamc@34 803 : forall (P : even_list -> Prop) (P0 : odd_list -> Prop),
adamc@34 804 P ENil ->
adamc@34 805 (forall (n : nat) (o : odd_list), P0 o -> P (ECons n o)) ->
adamc@34 806 (forall (n : nat) (e : even_list), P e -> P0 (OCons n e)) ->
adamc@34 807 forall e : even_list, P e
adamc@34 808 ]]
adamc@34 809
adamc@34 810 We see a mutually-recursive [fix], with the different functions separated by [with] in the same way that they would be separated by [and] in ML. A final [for] clause identifies which of the mutually-recursive functions should be the final value of the [fix] expression. Using this definition as a template, we can reimplement [even_list_mut] directly. *)
adamc@34 811
adamc@34 812 Section even_list_mut'.
adamc@34 813 (** First, we need the properties that we are proving. *)
adamc@34 814 Variable Peven : even_list -> Prop.
adamc@34 815 Variable Podd : odd_list -> Prop.
adamc@34 816
adamc@34 817 (** Next, we need proofs of the three cases. *)
adamc@38 818 Hypothesis ENil_case : Peven ENil.
adamc@38 819 Hypothesis ECons_case : forall (n : nat) (o : odd_list), Podd o -> Peven (ECons n o).
adamc@38 820 Hypothesis OCons_case : forall (n : nat) (e : even_list), Peven e -> Podd (OCons n e).
adamc@34 821
adamc@34 822 (** Finally, we define the recursive functions. *)
adamc@34 823 Fixpoint even_list_mut' (e : even_list) : Peven e :=
adamc@34 824 match e return (Peven e) with
adamc@34 825 | ENil => ENil_case
adamc@34 826 | ECons n o => ECons_case n (odd_list_mut' o)
adamc@34 827 end
adamc@34 828 with odd_list_mut' (o : odd_list) : Podd o :=
adamc@34 829 match o return (Podd o) with
adamc@34 830 | OCons n e => OCons_case n (even_list_mut' e)
adamc@34 831 end.
adamc@34 832 End even_list_mut'.
adamc@34 833
adamc@34 834 (** Even induction principles for reflexive types are easy to implement directly. For our [formula] type, we can use a recursive definition much like those we wrote above. *)
adamc@34 835
adamc@34 836 Section formula_ind'.
adamc@34 837 Variable P : formula -> Prop.
adamc@38 838 Hypothesis Eq_case : forall n1 n2 : nat, P (Eq n1 n2).
adamc@38 839 Hypothesis And_case : forall f1 f2 : formula,
adamc@34 840 P f1 -> P f2 -> P (And f1 f2).
adamc@38 841 Hypothesis Forall_case : forall f : nat -> formula,
adamc@34 842 (forall n : nat, P (f n)) -> P (Forall f).
adamc@34 843
adamc@34 844 Fixpoint formula_ind' (f : formula) : P f :=
adamc@34 845 match f return (P f) with
adamc@34 846 | Eq n1 n2 => Eq_case n1 n2
adamc@34 847 | And f1 f2 => And_case (formula_ind' f1) (formula_ind' f2)
adamc@34 848 | Forall f' => Forall_case f' (fun n => formula_ind' (f' n))
adamc@34 849 end.
adamc@34 850 End formula_ind'.
adamc@34 851
adamc@35 852
adamc@35 853 (** * Nested Inductive Types *)
adamc@35 854
adamc@35 855 (** Suppose we want to extend our earlier type of binary trees to trees with arbitrary finite branching. We can use lists to give a simple definition. *)
adamc@35 856
adamc@35 857 Inductive nat_tree : Set :=
adamc@35 858 | NLeaf' : nat_tree
adamc@35 859 | NNode' : nat -> list nat_tree -> nat_tree.
adamc@35 860
adamc@35 861 (** This is an example of a %\textit{%#<i>#nested#</i>#%}% inductive type definition, because we use the type we are defining as an argument to a parametrized type family. Coq will not allow all such definitions; it effectively pretends that we are defining [nat_tree] mutually with a version of [list] specialized to [nat_tree], checking that the resulting expanded definition satisfies the usual rules. For instance, if we replaced [list] with a type family that used its parameter as a function argument, then the definition would be rejected as violating the positivity restriction.
adamc@35 862
adamc@35 863 Like we encountered for mutual inductive types, we find that the automatically-generated induction principle for [nat_tree] is too weak. *)
adamc@35 864
adamc@35 865 Check nat_tree_ind.
adamc@35 866 (** [[
adamc@35 867
adamc@35 868 nat_tree_ind
adamc@35 869 : forall P : nat_tree -> Prop,
adamc@35 870 P NLeaf' ->
adamc@35 871 (forall (n : nat) (l : list nat_tree), P (NNode' n l)) ->
adamc@35 872 forall n : nat_tree, P n
adamc@35 873 ]]
adamc@35 874
adamc@35 875 There is no command like [Scheme] that will implement an improved principle for us. In general, it takes creativity to figure out how to incorporate nested uses to different type families. Now that we know how to implement induction principles manually, we are in a position to apply just such creativity to this problem.
adamc@35 876
adamc@35 877 First, we will need an auxiliary definition, characterizing what it means for a property to hold of every element of a list. *)
adamc@35 878
adamc@35 879 Section All.
adamc@35 880 Variable T : Set.
adamc@35 881 Variable P : T -> Prop.
adamc@35 882
adamc@35 883 Fixpoint All (ls : list T) : Prop :=
adamc@35 884 match ls with
adamc@35 885 | Nil => True
adamc@35 886 | Cons h t => P h /\ All t
adamc@35 887 end.
adamc@35 888 End All.
adamc@35 889
adamc@35 890 (** It will be useful to look at the definitions of [True] and [/\], since we will want to write manual proofs of them below. *)
adamc@35 891
adamc@35 892 Print True.
adamc@35 893 (** [[
adamc@35 894
adamc@35 895 Inductive True : Prop := I : True
adamc@35 896 ]]
adamc@35 897
adamc@35 898 That is, [True] is a proposition with exactly one proof, [I], which we may always supply trivially.
adamc@35 899
adamc@35 900 Finding the definition of [/\] takes a little more work. Coq supports user registration of arbitrary parsing rules, and it is such a rule that is letting us write [/\] instead of an application of some inductive type family. We can find the underlying inductive type with the [Locate] command. *)
adamc@35 901
adamc@35 902 Locate "/\".
adamc@35 903 (** [[
adamc@35 904
adamc@35 905 Notation Scope
adamc@35 906 "A /\ B" := and A B : type_scope
adamc@35 907 (default interpretation)
adamc@35 908 ]] *)
adamc@35 909
adamc@35 910 Print and.
adamc@35 911 (** [[
adamc@35 912
adamc@35 913 Inductive and (A : Prop) (B : Prop) : Prop := conj : A -> B -> A /\ B
adamc@35 914 For conj: Arguments A, B are implicit
adamc@35 915 For and: Argument scopes are [type_scope type_scope]
adamc@35 916 For conj: Argument scopes are [type_scope type_scope _ _]
adamc@35 917 ]]
adamc@35 918
adamc@35 919 In addition to the definition of [and] itself, we get information on implicit arguments and parsing rules for [and] and its constructor [conj]. We will ignore the parsing information for now. The implicit argument information tells us that we build a proof of a conjunction by calling the constructor [conj] on proofs of the conjuncts, with no need to include the types of those proofs as explicit arguments.
adamc@35 920
adamc@35 921 %\medskip%
adamc@35 922
adamc@35 923 Now we create a section for our induction principle, following the same basic plan as in the last section of this chapter. *)
adamc@35 924
adamc@35 925 Section nat_tree_ind'.
adamc@35 926 Variable P : nat_tree -> Prop.
adamc@35 927
adamc@38 928 Hypothesis NLeaf'_case : P NLeaf'.
adamc@38 929 Hypothesis NNode'_case : forall (n : nat) (ls : list nat_tree),
adamc@35 930 All P ls -> P (NNode' n ls).
adamc@35 931
adamc@35 932 (** A first attempt at writing the induction principle itself follows the intuition that nested inductive type definitions are expanded into mutual inductive definitions.
adamc@35 933
adamc@35 934 [[
adamc@35 935
adamc@35 936 Fixpoint nat_tree_ind' (tr : nat_tree) : P tr :=
adamc@35 937 match tr return (P tr) with
adamc@35 938 | NLeaf' => NLeaf'_case
adamc@35 939 | NNode' n ls => NNode'_case n ls (list_nat_tree_ind ls)
adamc@35 940 end
adamc@35 941
adamc@35 942 with list_nat_tree_ind (ls : list nat_tree) : All P ls :=
adamc@35 943 match ls return (All P ls) with
adamc@35 944 | Nil => I
adamc@35 945 | Cons tr rest => conj (nat_tree_ind' tr) (list_nat_tree_ind rest)
adamc@35 946 end.
adamc@35 947
adamc@35 948 Coq rejects this definition, saying "Recursive call to nat_tree_ind' has principal argument equal to "tr" instead of rest." The term "nested inductive type" hints at the solution to the problem. Just like true mutually-inductive types require mutually-recursive induction principles, nested types require nested recursion. *)
adamc@35 949
adamc@35 950 Fixpoint nat_tree_ind' (tr : nat_tree) : P tr :=
adamc@35 951 match tr return (P tr) with
adamc@35 952 | NLeaf' => NLeaf'_case
adamc@35 953 | NNode' n ls => NNode'_case n ls
adamc@35 954 ((fix list_nat_tree_ind (ls : list nat_tree) : All P ls :=
adamc@35 955 match ls return (All P ls) with
adamc@35 956 | Nil => I
adamc@35 957 | Cons tr rest => conj (nat_tree_ind' tr) (list_nat_tree_ind rest)
adamc@35 958 end) ls)
adamc@35 959 end.
adamc@35 960
adamc@35 961 (** We include an anonymous [fix] version of [list_nat_tree_ind] that is literally %\textit{%#<i>#nested#</i>#%}% inside the definition of the recursive function corresponding to the inductive definition that had the nested use of [list]. *)
adamc@35 962
adamc@35 963 End nat_tree_ind'.
adamc@35 964
adamc@35 965 (** We can try our induction principle out by defining some recursive functions on [nat_tree]s and proving a theorem about them. First, we define some helper functions that operate on lists. *)
adamc@35 966
adamc@35 967 Section map.
adamc@35 968 Variables T T' : Set.
adamc@35 969 Variable f : T -> T'.
adamc@35 970
adamc@35 971 Fixpoint map (ls : list T) : list T' :=
adamc@35 972 match ls with
adamc@35 973 | Nil => Nil
adamc@35 974 | Cons h t => Cons (f h) (map t)
adamc@35 975 end.
adamc@35 976 End map.
adamc@35 977
adamc@35 978 Fixpoint sum (ls : list nat) : nat :=
adamc@35 979 match ls with
adamc@35 980 | Nil => O
adamc@35 981 | Cons h t => plus h (sum t)
adamc@35 982 end.
adamc@35 983
adamc@35 984 (** Now we can define a size function over our trees. *)
adamc@35 985
adamc@35 986 Fixpoint ntsize (tr : nat_tree) : nat :=
adamc@35 987 match tr with
adamc@35 988 | NLeaf' => S O
adamc@35 989 | NNode' _ trs => S (sum (map ntsize trs))
adamc@35 990 end.
adamc@35 991
adamc@35 992 (** Notice that Coq was smart enough to expand the definition of [map] to verify that we are using proper nested recursion, even through a use of a higher-order function. *)
adamc@35 993
adamc@35 994 Fixpoint ntsplice (tr1 tr2 : nat_tree) {struct tr1} : nat_tree :=
adamc@35 995 match tr1 with
adamc@35 996 | NLeaf' => NNode' O (Cons tr2 Nil)
adamc@35 997 | NNode' n Nil => NNode' n (Cons tr2 Nil)
adamc@35 998 | NNode' n (Cons tr trs) => NNode' n (Cons (ntsplice tr tr2) trs)
adamc@35 999 end.
adamc@35 1000
adamc@35 1001 (** We have defined another arbitrary notion of tree splicing, similar to before, and we can prove an analogous theorem about its relationship with tree size. We start with a useful lemma about addition. *)
adamc@35 1002
adamc@41 1003 (* begin thide *)
adamc@35 1004 Lemma plus_S : forall n1 n2 : nat,
adamc@35 1005 plus n1 (S n2) = S (plus n1 n2).
adamc@35 1006 induction n1; crush.
adamc@35 1007 Qed.
adamc@41 1008 (* end thide *)
adamc@35 1009
adamc@35 1010 (** Now we begin the proof of the theorem, adding the lemma [plus_S] as a hint. *)
adamc@35 1011
adamc@35 1012 Theorem ntsize_ntsplice : forall tr1 tr2 : nat_tree, ntsize (ntsplice tr1 tr2)
adamc@35 1013 = plus (ntsize tr2) (ntsize tr1).
adamc@41 1014 (* begin thide *)
adamc@35 1015 Hint Rewrite plus_S : cpdt.
adamc@35 1016
adamc@35 1017 (** We know that the standard induction principle is insufficient for the task, so we need to provide a [using] clause for the [induction] tactic to specify our alternate principle. *)
adamc@35 1018 induction tr1 using nat_tree_ind'; crush.
adamc@35 1019
adamc@35 1020 (** One subgoal remains: [[
adamc@35 1021
adamc@35 1022 n : nat
adamc@35 1023 ls : list nat_tree
adamc@35 1024 H : All
adamc@35 1025 (fun tr1 : nat_tree =>
adamc@35 1026 forall tr2 : nat_tree,
adamc@35 1027 ntsize (ntsplice tr1 tr2) = plus (ntsize tr2) (ntsize tr1)) ls
adamc@35 1028 tr2 : nat_tree
adamc@35 1029 ============================
adamc@35 1030 ntsize
adamc@35 1031 match ls with
adamc@35 1032 | Nil => NNode' n (Cons tr2 Nil)
adamc@35 1033 | Cons tr trs => NNode' n (Cons (ntsplice tr tr2) trs)
adamc@35 1034 end = S (plus (ntsize tr2) (sum (map ntsize ls)))
adamc@35 1035 ]]
adamc@35 1036
adamc@35 1037 After a few moments of squinting at this goal, it becomes apparent that we need to do a case analysis on the structure of [ls]. The rest is routine. *)
adamc@35 1038
adamc@35 1039 destruct ls; crush.
adamc@35 1040
adamc@36 1041 (** We can go further in automating the proof by exploiting the hint mechanism. *)
adamc@35 1042
adamc@35 1043 Restart.
adamc@35 1044 Hint Extern 1 (ntsize (match ?LS with Nil => _ | Cons _ _ => _ end) = _) =>
adamc@35 1045 destruct LS; crush.
adamc@35 1046 induction tr1 using nat_tree_ind'; crush.
adamc@35 1047 Qed.
adamc@41 1048 (* end thide *)
adamc@35 1049
adamc@35 1050 (** We will go into great detail on hints in a later chapter, but the only important thing to note here is that we register a pattern that describes a conclusion we expect to encounter during the proof. The pattern may contain unification variables, whose names are prefixed with question marks, and we may refer to those bound variables in a tactic that we ask to have run whenever the pattern matches.
adamc@35 1051
adamc@40 1052 The advantage of using the hint is not very clear here, because the original proof was so short. However, the hint has fundamentally improved the readability of our proof. Before, the proof referred to the local variable [ls], which has an automatically-generated name. To a human reading the proof script without stepping through it interactively, it was not clear where [ls] came from. The hint explains to the reader the process for choosing which variables to case analyze on, and the hint can continue working even if the rest of the proof structure changes significantly. *)
adamc@36 1053
adamc@36 1054
adamc@36 1055 (** * Manual Proofs About Constructors *)
adamc@36 1056
adamc@36 1057 (** It can be useful to understand how tactics like [discriminate] and [injection] work, so it is worth stepping through a manual proof of each kind. We will start with a proof fit for [discriminate]. *)
adamc@36 1058
adamc@36 1059 Theorem true_neq_false : true <> false.
adamc@41 1060 (* begin thide *)
adamc@36 1061 (** We begin with the tactic [red], which is short for "one step of reduction," to unfold the definition of logical negation. *)
adamc@36 1062
adamc@36 1063 red.
adamc@36 1064 (** [[
adamc@36 1065
adamc@36 1066 ============================
adamc@36 1067 true = false -> False
adamc@36 1068 ]]
adamc@36 1069
adamc@36 1070 The negation is replaced with an implication of falsehood. We use the tactic [intro H] to change the assumption of the implication into a hypothesis named [H]. *)
adamc@36 1071
adamc@36 1072 intro H.
adamc@36 1073 (** [[
adamc@36 1074
adamc@36 1075 H : true = false
adamc@36 1076 ============================
adamc@36 1077 False
adamc@36 1078 ]]
adamc@36 1079
adamc@36 1080 This is the point in the proof where we apply some creativity. We define a function whose utility will become clear soon. *)
adamc@36 1081
adamc@36 1082 Definition f (b : bool) := if b then True else False.
adamc@36 1083
adamc@36 1084 (** It is worth recalling the difference between the lowercase and uppercase versions of truth and falsehood: [True] and [False] are logical propositions, while [true] and [false] are boolean values that we can case-analyze. We have defined [f] such that our conclusion of [False] is computationally equivalent to [f false]. Thus, the [change] tactic will let us change the conclusion to [f false]. *)
adamc@36 1085
adamc@36 1086 change (f false).
adamc@36 1087 (** [[
adamc@36 1088
adamc@36 1089 H : true = false
adamc@36 1090 ============================
adamc@36 1091 f false
adamc@36 1092 ]]
adamc@36 1093
adamc@36 1094 Now the righthand side of [H]'s equality appears in the conclusion, so we can rewrite. *)
adamc@36 1095
adamc@36 1096 rewrite <- H.
adamc@36 1097 (** [[
adamc@36 1098
adamc@36 1099 H : true = false
adamc@36 1100 ============================
adamc@36 1101 f true
adamc@36 1102 ]]
adamc@36 1103
adamc@36 1104 We are almost done. Just how close we are to done is revealed by computational simplification. *)
adamc@36 1105
adamc@36 1106 simpl.
adamc@36 1107 (** [[
adamc@36 1108
adamc@36 1109 H : true = false
adamc@36 1110 ============================
adamc@36 1111 True
adamc@36 1112 ]] *)
adamc@36 1113
adamc@36 1114 trivial.
adamc@36 1115 Qed.
adamc@41 1116 (* end thide *)
adamc@36 1117
adamc@36 1118 (** I have no trivial automated version of this proof to suggest, beyond using [discriminate] or [congruence] in the first place.
adamc@36 1119
adamc@36 1120 %\medskip%
adamc@36 1121
adamc@36 1122 We can perform a similar manual proof of injectivity of the constructor [S]. I leave a walk-through of the details to curious readers who want to run the proof script interactively. *)
adamc@36 1123
adamc@36 1124 Theorem S_inj' : forall n m : nat, S n = S m -> n = m.
adamc@41 1125 (* begin thide *)
adamc@36 1126 intros n m H.
adamc@36 1127 change (pred (S n) = pred (S m)).
adamc@36 1128 rewrite H.
adamc@36 1129 reflexivity.
adamc@36 1130 Qed.
adamc@41 1131 (* end thide *)
adamc@36 1132
adamc@37 1133
adamc@37 1134 (** * Exercises *)
adamc@37 1135
adamc@37 1136 (** %\begin{enumerate}%#<ol>#
adamc@37 1137
adamc@37 1138 %\item%#<li># Define an inductive type [truth] with three constructors, [Yes], [No], and [Maybe]. [Yes] stands for certain truth, [False] for certain falsehood, and [Maybe] for an unknown situation. Define "not," "and," and "or" for this replacement boolean algebra. Prove that your implementation of "and" is commutative and distributes over your implementation of "or."#</li>#
adamc@37 1139
adamc@39 1140 %\item%#<li># Modify the first example language of Chapter 2 to include variables, where variables are represented with [nat]. Extend the syntax and semantics of expressions to accommodate the change. Your new [expDenote] function should take as a new extra first argument a value of type [var -> nat], where [var] is a synonym for naturals-as-variables, and the function assigns a value to each variable. Define a constant folding function which does a bottom-up pass over an expression, at each stage replacing every binary operation on constants with an equivalent constant. Prove that constant folding preserves the meanings of expressions.#</li>#
adamc@38 1141
adamc@39 1142 %\item%#<li># Reimplement the second example language of Chapter 2 to use mutually-inductive types instead of dependent types. That is, define two separate (non-dependent) inductive types [nat_exp] and [bool_exp] for expressions of the two different types, rather than a single indexed type. To keep things simple, you may consider only the binary operators that take naturals as operands. Add natural number variables to the language, as in the last exercise, and add an "if" expression form taking as arguments one boolean expression and two natural number expressions. Define semantics and constant-folding functions for this new language. Your constant folding should simplify not just binary operations (returning naturals or booleans) with known arguments, but also "if" expressions with known values for their test expressions but possibly undetermined "then" and "else" cases. Prove that constant-folding a natural number expression preserves its meaning.#</li>#
adamc@38 1143
adamc@38 1144 %\item%#<li># Using a reflexive inductive definition, define a type [nat_tree] of infinitary trees, with natural numbers at their leaves and a countable infinity of new trees branching out of each internal node. Define a function [increment] that increments the number in every leaf of a [nat_tree]. Define a function [leapfrog] over a natural [i] and a tree [nt]. [leapfrog] should recurse into the [i]th child of [nt], the [i+1]st child of that node, the [i+2]nd child of the next node, and so on, until reaching a leaf, in which case [leapfrog] should return the number at that leaf. Prove that the result of any call to [leapfrog] is incremented by one by calling [increment] on the tree.#</li>#
adamc@38 1145
adamc@38 1146 %\item%#<li># Define a type of trees of trees of trees of (repeat to infinity). That is, define an inductive type [trexp], whose members are either base cases containing natural numbers or binary trees of [trexp]s. Base your definition on a parameterized binary tree type [btree] that you will also define, so that [trexp] is defined as a nested inductive type. Define a function [total] that sums all of the naturals at the leaves of a [trexp]. Define a function [increment] that increments every leaf of a [trexp] by one. Prove that, for all [tr], [total (increment tr) >= total tr]. On the way to finishing this proof, you will probably want to prove a lemma and add it as a hint using the syntax [Hint Resolve name_of_lemma.].#</li>#
adamc@38 1147
adamc@38 1148 %\item%#<li># Prove discrimination and injectivity theorems for the [nat_btree] type defined earlier in this chapter. In particular, without using the tactics [discriminate], [injection], or [congruence], prove that no leaf equals any node, and prove that two equal nodes carry the same natural number.#</li>#
adamc@37 1149
adamc@37 1150 #</ol>#%\end{enumerate}% *)