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author Adam Chlipala <adamc@hcoop.net>
date Fri, 28 Nov 2008 14:10:23 -0500
parents a21447f76aad
children a2b14ba218a7
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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@74 19 (** %\part{Basic Programming and Proving}
adamc@74 20
adamc@74 21 \chapter{Introducing Inductive Types}% *)
adamc@26 22
adamc@45 23 (** 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 24
adamc@26 25
adamc@26 26 (** * Enumerations *)
adamc@26 27
adamc@26 28 (** 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 29
adamc@26 30 The singleton type [unit] is an inductive type: *)
adamc@26 31
adamc@26 32 Inductive unit : Set :=
adamc@26 33 | tt.
adamc@26 34
adamc@26 35 (** 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 36
adamc@26 37 Check unit.
adamc@26 38 (** [[
adamc@26 39
adamc@26 40 unit : Set
adamc@26 41 ]] *)
adamc@26 42 Check tt.
adamc@26 43 (** [[
adamc@26 44
adamc@26 45 tt : unit
adamc@26 46 ]] *)
adamc@26 47
adamc@26 48 (** We can prove that [unit] is a genuine singleton type. *)
adamc@26 49
adamc@26 50 Theorem unit_singleton : forall x : unit, x = tt.
adamc@26 51 (** 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 52 (* begin thide *)
adamc@26 53 induction x.
adamc@26 54 (** The goal changes to: [[
adamc@26 55
adamc@26 56 tt = tt
adamc@26 57 ]] *)
adamc@26 58 (** ...which we can discharge trivially. *)
adamc@26 59 reflexivity.
adamc@26 60 Qed.
adamc@41 61 (* end thide *)
adamc@26 62
adamc@26 63 (** 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 64
adamc@26 65 destruct x.
adamc@26 66 ...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 67
adamc@26 68 What exactly %\textit{%#<i>#is#</i>#%}% the induction principle for [unit]? We can ask Coq: *)
adamc@26 69
adamc@26 70 Check unit_ind.
adamc@26 71 (** [[
adamc@26 72
adamc@26 73 unit_ind : forall P : unit -> Prop, P tt -> forall u : unit, P u
adamc@26 74 ]]
adamc@26 75
adamc@26 76 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 77
adamc@26 78 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 79
adamc@26 80 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 81
adamc@26 82 %\medskip%
adamc@26 83
adamc@26 84 We can define an inductive type even simpler than [unit]: *)
adamc@26 85
adamc@26 86 Inductive Empty_set : Set := .
adamc@26 87
adamc@26 88 (** [Empty_set] has no elements. We can prove fun theorems about it: *)
adamc@26 89
adamc@26 90 Theorem the_sky_is_falling : forall x : Empty_set, 2 + 2 = 5.
adamc@41 91 (* begin thide *)
adamc@26 92 destruct 1.
adamc@26 93 Qed.
adamc@41 94 (* end thide *)
adamc@26 95
adamc@32 96 (** 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 97
adamc@26 98 We can see the induction principle that made this proof so easy: *)
adamc@26 99
adamc@26 100 Check Empty_set_ind.
adamc@26 101 (** [[
adamc@26 102
adamc@26 103 Empty_set_ind : forall (P : Empty_set -> Prop) (e : Empty_set), P e
adamc@26 104 ]]
adamc@26 105
adamc@26 106 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@26 107
adamc@26 108 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@26 109
adamc@26 110 Definition e2u (e : Empty_set) : unit := match e with end.
adamc@26 111
adamc@26 112 (** 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@26 113
adamc@26 114 %\medskip%
adamc@26 115
adamc@26 116 Moving up the ladder of complexity, we can define the booleans: *)
adamc@26 117
adamc@26 118 Inductive bool : Set :=
adamc@26 119 | true
adamc@26 120 | false.
adamc@26 121
adamc@26 122 (** We can use less vacuous pattern matching to define boolean negation. *)
adamc@26 123
adamc@26 124 Definition not (b : bool) : bool :=
adamc@26 125 match b with
adamc@26 126 | true => false
adamc@26 127 | false => true
adamc@26 128 end.
adamc@26 129
adamc@27 130 (** An alternative definition desugars to the above: *)
adamc@27 131
adamc@27 132 Definition not' (b : bool) : bool :=
adamc@27 133 if b then false else true.
adamc@27 134
adamc@26 135 (** We might want to prove that [not] is its own inverse operation. *)
adamc@26 136
adamc@26 137 Theorem not_inverse : forall b : bool, not (not b) = b.
adamc@41 138 (* begin thide *)
adamc@26 139 destruct b.
adamc@26 140
adamc@26 141 (** After we case analyze on [b], we are left with one subgoal for each constructor of [bool].
adamc@26 142
adamc@26 143 [[
adamc@26 144
adamc@26 145 2 subgoals
adamc@26 146
adamc@26 147 ============================
adamc@26 148 not (not true) = true
adamc@26 149 ]]
adamc@26 150
adamc@26 151 [[
adamc@26 152 subgoal 2 is:
adamc@26 153 not (not false) = false
adamc@26 154 ]]
adamc@26 155
adamc@26 156 The first subgoal follows by Coq's rules of computation, so we can dispatch it easily: *)
adamc@26 157
adamc@26 158 reflexivity.
adamc@26 159
adamc@26 160 (** Likewise for the second subgoal, so we can restart the proof and give a very compact justification. *)
adamc@26 161
adamc@26 162 Restart.
adamc@26 163 destruct b; reflexivity.
adamc@26 164 Qed.
adamc@41 165 (* end thide *)
adamc@27 166
adamc@27 167 (** Another theorem about booleans illustrates another useful tactic. *)
adamc@27 168
adamc@27 169 Theorem not_ineq : forall b : bool, not b <> b.
adamc@41 170 (* begin thide *)
adamc@27 171 destruct b; discriminate.
adamc@27 172 Qed.
adamc@41 173 (* end thide *)
adamc@27 174
adamc@27 175 (** [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@27 176
adamc@27 177 At this point, it is probably not hard to guess what the underlying induction principle for [bool] is. *)
adamc@27 178
adamc@27 179 Check bool_ind.
adamc@27 180 (** [[
adamc@27 181
adamc@27 182 bool_ind : forall P : bool -> Prop, P true -> P false -> forall b : bool, P b
adamc@27 183 ]] *)
adamc@28 184
adamc@28 185
adamc@28 186 (** * Simple Recursive Types *)
adamc@28 187
adamc@28 188 (** The natural numbers are the simplest common example of an inductive type that actually deserves the name. *)
adamc@28 189
adamc@28 190 Inductive nat : Set :=
adamc@28 191 | O : nat
adamc@28 192 | S : nat -> nat.
adamc@28 193
adamc@28 194 (** [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@28 195
adamc@28 196 Pattern matching works as we demonstrated in the last chapter: *)
adamc@28 197
adamc@28 198 Definition isZero (n : nat) : bool :=
adamc@28 199 match n with
adamc@28 200 | O => true
adamc@28 201 | S _ => false
adamc@28 202 end.
adamc@28 203
adamc@28 204 Definition pred (n : nat) : nat :=
adamc@28 205 match n with
adamc@28 206 | O => O
adamc@28 207 | S n' => n'
adamc@28 208 end.
adamc@28 209
adamc@28 210 (** We can prove theorems by case analysis: *)
adamc@28 211
adamc@28 212 Theorem S_isZero : forall n : nat, isZero (pred (S (S n))) = false.
adamc@41 213 (* begin thide *)
adamc@28 214 destruct n; reflexivity.
adamc@28 215 Qed.
adamc@41 216 (* end thide *)
adamc@28 217
adamc@28 218 (** We can also now get into genuine inductive theorems. First, we will need a recursive function, to make things interesting. *)
adamc@28 219
adamc@28 220 Fixpoint plus (n m : nat) {struct n} : nat :=
adamc@28 221 match n with
adamc@28 222 | O => m
adamc@28 223 | S n' => S (plus n' m)
adamc@28 224 end.
adamc@28 225
adamc@28 226 (** 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@28 227
adamc@28 228 Some theorems about [plus] can be proved without induction. *)
adamc@28 229
adamc@28 230 Theorem O_plus_n : forall n : nat, plus O n = n.
adamc@41 231 (* begin thide *)
adamc@28 232 intro; reflexivity.
adamc@28 233 Qed.
adamc@41 234 (* end thide *)
adamc@28 235
adamc@28 236 (** 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@28 237
adamc@28 238 Theorem n_plus_O : forall n : nat, plus n O = n.
adamc@41 239 (* begin thide *)
adamc@28 240 induction n.
adamc@28 241
adamc@28 242 (** Our first subgoal is [plus O O = O], which %\textit{%#<i>#is#</i>#%}% trivial by computation. *)
adamc@28 243
adamc@28 244 reflexivity.
adamc@28 245
adamc@28 246 (** Our second subgoal is more work and also demonstrates our first inductive hypothesis.
adamc@28 247
adamc@28 248 [[
adamc@28 249
adamc@28 250 n : nat
adamc@28 251 IHn : plus n O = n
adamc@28 252 ============================
adamc@28 253 plus (S n) O = S n
adamc@28 254 ]]
adamc@28 255
adamc@28 256 We can start out by using computation to simplify the goal as far as we can. *)
adamc@28 257
adamc@28 258 simpl.
adamc@28 259
adamc@28 260 (** Now the conclusion is [S (plus n O) = S n]. Using our inductive hypothesis: *)
adamc@28 261
adamc@28 262 rewrite IHn.
adamc@28 263
adamc@28 264 (** ...we get a trivial conclusion [S n = S n]. *)
adamc@28 265
adamc@28 266 reflexivity.
adamc@28 267
adamc@28 268 (** Not much really went on in this proof, so the [crush] tactic from the [Tactics] module can prove this theorem automatically. *)
adamc@28 269
adamc@28 270 Restart.
adamc@28 271 induction n; crush.
adamc@28 272 Qed.
adamc@41 273 (* end thide *)
adamc@28 274
adamc@28 275 (** We can check out the induction principle at work here: *)
adamc@28 276
adamc@28 277 Check nat_ind.
adamc@28 278 (** [[
adamc@28 279
adamc@28 280 nat_ind : forall P : nat -> Prop,
adamc@28 281 P O -> (forall n : nat, P n -> P (S n)) -> forall n : nat, P n
adamc@28 282 ]]
adamc@28 283
adamc@28 284 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@28 285
adamc@28 286 Since [nat] has a constructor that takes an argument, we may sometimes need to know that that constructor is injective. *)
adamc@28 287
adamc@28 288 Theorem S_inj : forall n m : nat, S n = S m -> n = m.
adamc@41 289 (* begin thide *)
adamc@28 290 injection 1; trivial.
adamc@28 291 Qed.
adamc@41 292 (* end thide *)
adamc@28 293
adamc@28 294 (** [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@28 295
adamc@29 296 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@29 297
adamc@29 298 %\medskip%
adamc@29 299
adamc@29 300 We can define a type of lists of natural numbers. *)
adamc@29 301
adamc@29 302 Inductive nat_list : Set :=
adamc@29 303 | NNil : nat_list
adamc@29 304 | NCons : nat -> nat_list -> nat_list.
adamc@29 305
adamc@29 306 (** Recursive definitions are straightforward extensions of what we have seen before. *)
adamc@29 307
adamc@29 308 Fixpoint nlength (ls : nat_list) : nat :=
adamc@29 309 match ls with
adamc@29 310 | NNil => O
adamc@29 311 | NCons _ ls' => S (nlength ls')
adamc@29 312 end.
adamc@29 313
adamc@29 314 Fixpoint napp (ls1 ls2 : nat_list) {struct ls1} : nat_list :=
adamc@29 315 match ls1 with
adamc@29 316 | NNil => ls2
adamc@29 317 | NCons n ls1' => NCons n (napp ls1' ls2)
adamc@29 318 end.
adamc@29 319
adamc@29 320 (** Inductive theorem proving can again be automated quite effectively. *)
adamc@29 321
adamc@29 322 Theorem nlength_napp : forall ls1 ls2 : nat_list, nlength (napp ls1 ls2)
adamc@29 323 = plus (nlength ls1) (nlength ls2).
adamc@41 324 (* begin thide *)
adamc@29 325 induction ls1; crush.
adamc@29 326 Qed.
adamc@41 327 (* end thide *)
adamc@29 328
adamc@29 329 Check nat_list_ind.
adamc@29 330 (** [[
adamc@29 331
adamc@29 332 nat_list_ind
adamc@29 333 : forall P : nat_list -> Prop,
adamc@29 334 P NNil ->
adamc@29 335 (forall (n : nat) (n0 : nat_list), P n0 -> P (NCons n n0)) ->
adamc@29 336 forall n : nat_list, P n
adamc@29 337 ]]
adamc@29 338
adamc@29 339 %\medskip%
adamc@29 340
adamc@29 341 In general, we can implement any "tree" types as inductive types. For example, here are binary trees of naturals. *)
adamc@29 342
adamc@29 343 Inductive nat_btree : Set :=
adamc@29 344 | NLeaf : nat_btree
adamc@29 345 | NNode : nat_btree -> nat -> nat_btree -> nat_btree.
adamc@29 346
adamc@29 347 Fixpoint nsize (tr : nat_btree) : nat :=
adamc@29 348 match tr with
adamc@35 349 | NLeaf => S O
adamc@29 350 | NNode tr1 _ tr2 => plus (nsize tr1) (nsize tr2)
adamc@29 351 end.
adamc@29 352
adamc@29 353 Fixpoint nsplice (tr1 tr2 : nat_btree) {struct tr1} : nat_btree :=
adamc@29 354 match tr1 with
adamc@35 355 | NLeaf => NNode tr2 O NLeaf
adamc@29 356 | NNode tr1' n tr2' => NNode (nsplice tr1' tr2) n tr2'
adamc@29 357 end.
adamc@29 358
adamc@29 359 Theorem plus_assoc : forall n1 n2 n3 : nat, plus (plus n1 n2) n3 = plus n1 (plus n2 n3).
adamc@41 360 (* begin thide *)
adamc@29 361 induction n1; crush.
adamc@29 362 Qed.
adamc@41 363 (* end thide *)
adamc@29 364
adamc@29 365 Theorem nsize_nsplice : forall tr1 tr2 : nat_btree, nsize (nsplice tr1 tr2)
adamc@29 366 = plus (nsize tr2) (nsize tr1).
adamc@41 367 (* begin thide *)
adamc@29 368 Hint Rewrite n_plus_O plus_assoc : cpdt.
adamc@29 369
adamc@29 370 induction tr1; crush.
adamc@29 371 Qed.
adamc@41 372 (* end thide *)
adamc@29 373
adamc@29 374 Check nat_btree_ind.
adamc@29 375 (** [[
adamc@29 376
adamc@29 377 nat_btree_ind
adamc@29 378 : forall P : nat_btree -> Prop,
adamc@29 379 P NLeaf ->
adamc@29 380 (forall n : nat_btree,
adamc@29 381 P n -> forall (n0 : nat) (n1 : nat_btree), P n1 -> P (NNode n n0 n1)) ->
adamc@29 382 forall n : nat_btree, P n
adamc@29 383 ]] *)
adamc@30 384
adamc@30 385
adamc@30 386 (** * Parameterized Types *)
adamc@30 387
adamc@30 388 (** We can also define polymorphic inductive types, as with algebraic datatypes in Haskell and ML. *)
adamc@30 389
adamc@30 390 Inductive list (T : Set) : Set :=
adamc@30 391 | Nil : list T
adamc@30 392 | Cons : T -> list T -> list T.
adamc@30 393
adamc@30 394 Fixpoint length T (ls : list T) : nat :=
adamc@30 395 match ls with
adamc@30 396 | Nil => O
adamc@30 397 | Cons _ ls' => S (length ls')
adamc@30 398 end.
adamc@30 399
adamc@30 400 Fixpoint app T (ls1 ls2 : list T) {struct ls1} : list T :=
adamc@30 401 match ls1 with
adamc@30 402 | Nil => ls2
adamc@30 403 | Cons x ls1' => Cons x (app ls1' ls2)
adamc@30 404 end.
adamc@30 405
adamc@30 406 Theorem length_app : forall T (ls1 ls2 : list T), length (app ls1 ls2)
adamc@30 407 = plus (length ls1) (length ls2).
adamc@41 408 (* begin thide *)
adamc@30 409 induction ls1; crush.
adamc@30 410 Qed.
adamc@41 411 (* end thide *)
adamc@30 412
adamc@30 413 (** 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@30 414
adamc@30 415 (* begin hide *)
adamc@30 416 Reset list.
adamc@30 417 (* end hide *)
adamc@30 418
adamc@30 419 Section list.
adamc@30 420 Variable T : Set.
adamc@30 421
adamc@30 422 Inductive list : Set :=
adamc@30 423 | Nil : list
adamc@30 424 | Cons : T -> list -> list.
adamc@30 425
adamc@30 426 Fixpoint length (ls : list) : nat :=
adamc@30 427 match ls with
adamc@30 428 | Nil => O
adamc@30 429 | Cons _ ls' => S (length ls')
adamc@30 430 end.
adamc@30 431
adamc@30 432 Fixpoint app (ls1 ls2 : list) {struct ls1} : list :=
adamc@30 433 match ls1 with
adamc@30 434 | Nil => ls2
adamc@30 435 | Cons x ls1' => Cons x (app ls1' ls2)
adamc@30 436 end.
adamc@30 437
adamc@30 438 Theorem length_app : forall ls1 ls2 : list, length (app ls1 ls2)
adamc@30 439 = plus (length ls1) (length ls2).
adamc@41 440 (* begin thide *)
adamc@30 441 induction ls1; crush.
adamc@30 442 Qed.
adamc@41 443 (* end thide *)
adamc@30 444 End list.
adamc@30 445
adamc@35 446 (* begin hide *)
adamc@35 447 Implicit Arguments Nil [T].
adamc@35 448 (* end hide *)
adamc@35 449
adamc@30 450 (** After we end the section, the [Variable]s we used are added as extra function parameters for each defined identifier, as needed. *)
adamc@30 451
adamc@30 452 Check list.
adamc@30 453 (** [[
adamc@30 454
adamc@30 455 list
adamc@30 456 : Set -> Set
adamc@30 457 ]] *)
adamc@30 458
adamc@30 459 Check Cons.
adamc@30 460 (** [[
adamc@30 461
adamc@30 462 Cons
adamc@30 463 : forall T : Set, T -> list T -> list T
adamc@30 464 ]] *)
adamc@30 465
adamc@30 466 Check length.
adamc@30 467 (** [[
adamc@30 468
adamc@30 469 length
adamc@30 470 : forall T : Set, list T -> nat
adamc@30 471 ]]
adamc@30 472
adamc@30 473 The extra parameter [T] is treated as a new argument to the induction principle, too. *)
adamc@30 474
adamc@30 475 Check list_ind.
adamc@30 476 (** [[
adamc@30 477
adamc@30 478 list_ind
adamc@30 479 : forall (T : Set) (P : list T -> Prop),
adamc@30 480 P (Nil T) ->
adamc@30 481 (forall (t : T) (l : list T), P l -> P (Cons t l)) ->
adamc@30 482 forall l : list T, P l
adamc@30 483 ]]
adamc@30 484
adamc@30 485 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@31 486
adamc@31 487
adamc@31 488 (** * Mutually Inductive Types *)
adamc@31 489
adamc@31 490 (** We can define inductive types that refer to each other: *)
adamc@31 491
adamc@31 492 Inductive even_list : Set :=
adamc@31 493 | ENil : even_list
adamc@31 494 | ECons : nat -> odd_list -> even_list
adamc@31 495
adamc@31 496 with odd_list : Set :=
adamc@31 497 | OCons : nat -> even_list -> odd_list.
adamc@31 498
adamc@31 499 Fixpoint elength (el : even_list) : nat :=
adamc@31 500 match el with
adamc@31 501 | ENil => O
adamc@31 502 | ECons _ ol => S (olength ol)
adamc@31 503 end
adamc@31 504
adamc@31 505 with olength (ol : odd_list) : nat :=
adamc@31 506 match ol with
adamc@31 507 | OCons _ el => S (elength el)
adamc@31 508 end.
adamc@31 509
adamc@31 510 Fixpoint eapp (el1 el2 : even_list) {struct el1} : even_list :=
adamc@31 511 match el1 with
adamc@31 512 | ENil => el2
adamc@31 513 | ECons n ol => ECons n (oapp ol el2)
adamc@31 514 end
adamc@31 515
adamc@31 516 with oapp (ol : odd_list) (el : even_list) {struct ol} : odd_list :=
adamc@31 517 match ol with
adamc@31 518 | OCons n el' => OCons n (eapp el' el)
adamc@31 519 end.
adamc@31 520
adamc@31 521 (** Everything is going roughly the same as in past examples, until we try to prove a theorem similar to those that came before. *)
adamc@31 522
adamc@31 523 Theorem elength_eapp : forall el1 el2 : even_list,
adamc@31 524 elength (eapp el1 el2) = plus (elength el1) (elength el2).
adamc@41 525 (* begin thide *)
adamc@31 526 induction el1; crush.
adamc@31 527
adamc@31 528 (** One goal remains: [[
adamc@31 529
adamc@31 530 n : nat
adamc@31 531 o : odd_list
adamc@31 532 el2 : even_list
adamc@31 533 ============================
adamc@31 534 S (olength (oapp o el2)) = S (plus (olength o) (elength el2))
adamc@31 535 ]]
adamc@31 536
adamc@31 537 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@31 538
adamc@31 539 Abort.
adamc@31 540 Check even_list_ind.
adamc@31 541 (** [[
adamc@31 542
adamc@31 543 even_list_ind
adamc@31 544 : forall P : even_list -> Prop,
adamc@31 545 P ENil ->
adamc@31 546 (forall (n : nat) (o : odd_list), P (ECons n o)) ->
adamc@31 547 forall e : even_list, P e
adamc@31 548 ]]
adamc@31 549
adamc@31 550 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@31 551
adamc@31 552 Scheme even_list_mut := Induction for even_list Sort Prop
adamc@31 553 with odd_list_mut := Induction for odd_list Sort Prop.
adamc@31 554
adamc@31 555 Check even_list_mut.
adamc@31 556 (** [[
adamc@31 557
adamc@31 558 even_list_mut
adamc@31 559 : forall (P : even_list -> Prop) (P0 : odd_list -> Prop),
adamc@31 560 P ENil ->
adamc@31 561 (forall (n : nat) (o : odd_list), P0 o -> P (ECons n o)) ->
adamc@31 562 (forall (n : nat) (e : even_list), P e -> P0 (OCons n e)) ->
adamc@31 563 forall e : even_list, P e
adamc@31 564 ]]
adamc@31 565
adamc@31 566 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@31 567
adamc@31 568 Theorem n_plus_O' : forall n : nat, plus n O = n.
adamc@31 569 apply (nat_ind (fun n => plus n O = n)); crush.
adamc@31 570 Qed.
adamc@31 571
adamc@31 572 (** 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@31 573
adamc@31 574 This technique generalizes to our mutual example: *)
adamc@31 575
adamc@31 576 Theorem elength_eapp : forall el1 el2 : even_list,
adamc@31 577 elength (eapp el1 el2) = plus (elength el1) (elength el2).
adamc@41 578
adamc@31 579 apply (even_list_mut
adamc@31 580 (fun el1 : even_list => forall el2 : even_list,
adamc@31 581 elength (eapp el1 el2) = plus (elength el1) (elength el2))
adamc@31 582 (fun ol : odd_list => forall el : even_list,
adamc@31 583 olength (oapp ol el) = plus (olength ol) (elength el))); crush.
adamc@31 584 Qed.
adamc@41 585 (* end thide *)
adamc@31 586
adamc@31 587 (** 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@33 588
adamc@33 589
adamc@33 590 (** * Reflexive Types *)
adamc@33 591
adamc@33 592 (** 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@33 593
adamc@33 594 Inductive formula : Set :=
adamc@33 595 | Eq : nat -> nat -> formula
adamc@33 596 | And : formula -> formula -> formula
adamc@33 597 | Forall : (nat -> formula) -> formula.
adamc@33 598
adamc@33 599 (** 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@33 600
adamc@33 601 Example forall_refl : formula := Forall (fun x => Eq x x).
adamc@33 602
adamc@33 603 (** We can write recursive functions over reflexive types quite naturally. Here is one translating our formulas into native Coq propositions. *)
adamc@33 604
adamc@33 605 Fixpoint formulaDenote (f : formula) : Prop :=
adamc@33 606 match f with
adamc@33 607 | Eq n1 n2 => n1 = n2
adamc@33 608 | And f1 f2 => formulaDenote f1 /\ formulaDenote f2
adamc@33 609 | Forall f' => forall n : nat, formulaDenote (f' n)
adamc@33 610 end.
adamc@33 611
adamc@33 612 (** We can also encode a trivial formula transformation that swaps the order of equality and conjunction operands. *)
adamc@33 613
adamc@33 614 Fixpoint swapper (f : formula) : formula :=
adamc@33 615 match f with
adamc@33 616 | Eq n1 n2 => Eq n2 n1
adamc@33 617 | And f1 f2 => And (swapper f2) (swapper f1)
adamc@33 618 | Forall f' => Forall (fun n => swapper (f' n))
adamc@33 619 end.
adamc@33 620
adamc@33 621 (** It is helpful to prove that this transformation does not make true formulas false. *)
adamc@33 622
adamc@33 623 Theorem swapper_preserves_truth : forall f, formulaDenote f -> formulaDenote (swapper f).
adamc@41 624 (* begin thide *)
adamc@33 625 induction f; crush.
adamc@33 626 Qed.
adamc@41 627 (* end thide *)
adamc@33 628
adamc@33 629 (** We can take a look at the induction principle behind this proof. *)
adamc@33 630
adamc@33 631 Check formula_ind.
adamc@33 632 (** [[
adamc@33 633
adamc@33 634 formula_ind
adamc@33 635 : forall P : formula -> Prop,
adamc@33 636 (forall n n0 : nat, P (Eq n n0)) ->
adamc@33 637 (forall f0 : formula,
adamc@33 638 P f0 -> forall f1 : formula, P f1 -> P (And f0 f1)) ->
adamc@33 639 (forall f1 : nat -> formula,
adamc@33 640 (forall n : nat, P (f1 n)) -> P (Forall f1)) ->
adamc@33 641 forall f2 : formula, P f2
adamc@33 642 ]] *)
adamc@33 643
adamc@33 644 (** 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@33 645
adamc@33 646 %\medskip%
adamc@33 647
adamc@33 648 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@33 649
adamc@33 650 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@33 651
adamc@33 652 (** [[
adamc@33 653
adamc@33 654 Inductive term : Set :=
adamc@33 655 | App : term -> term -> term
adamc@33 656 | Abs : (term -> term) -> term.
adamc@33 657
adamc@33 658 [[
adamc@33 659 Error: Non strictly positive occurrence of "term" in "(term -> term) -> term"
adamc@33 660 ]]
adamc@33 661
adamc@33 662 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@33 663
adamc@33 664 Why must Coq enforce this restriction? Imagine that our last definition had been accepted, allowing us to write this function:
adamc@33 665
adamc@33 666 [[
adamc@33 667 Definition uhoh (t : term) : term :=
adamc@33 668 match t with
adamc@33 669 | Abs f => f t
adamc@33 670 | _ => t
adamc@33 671 end.
adamc@33 672
adamc@33 673 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@33 674
adamc@33 675 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@33 676
adamc@33 677 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@34 678
adamc@34 679
adamc@34 680 (** * An Interlude on Proof Terms *)
adamc@34 681
adamc@34 682 (** 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@34 683
adamc@34 684 Print unit_ind.
adamc@34 685 (** [[
adamc@34 686
adamc@34 687 unit_ind =
adamc@34 688 fun P : unit -> Prop => unit_rect P
adamc@34 689 : forall P : unit -> Prop, P tt -> forall u : unit, P u
adamc@34 690 ]]
adamc@34 691
adamc@34 692 We see that this induction principle is defined in terms of a more general principle, [unit_rect]. *)
adamc@34 693
adamc@34 694 Check unit_rect.
adamc@34 695 (** [[
adamc@34 696
adamc@34 697 unit_rect
adamc@34 698 : forall P : unit -> Type, P tt -> forall u : unit, P u
adamc@34 699 ]]
adamc@34 700
adamc@34 701 [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@34 702
adamc@34 703 Print unit_rec.
adamc@34 704 (** [[
adamc@34 705
adamc@34 706 unit_rec =
adamc@34 707 fun P : unit -> Set => unit_rect P
adamc@34 708 : forall P : unit -> Set, P tt -> forall u : unit, P u
adamc@34 709 ]]
adamc@34 710
adamc@34 711 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@34 712
adamc@34 713 Definition always_O (u : unit) : nat :=
adamc@34 714 match u with
adamc@34 715 | tt => O
adamc@34 716 end.
adamc@34 717
adamc@34 718 Definition always_O' (u : unit) : nat :=
adamc@34 719 unit_rec (fun _ : unit => nat) O u.
adamc@34 720
adamc@34 721 (** 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@34 722
adamc@34 723 Print unit_rect.
adamc@34 724
adamc@34 725 (** [[
adamc@34 726
adamc@34 727 unit_rect =
adamc@34 728 fun (P : unit -> Type) (f : P tt) (u : unit) =>
adamc@34 729 match u as u0 return (P u0) with
adamc@34 730 | tt => f
adamc@34 731 end
adamc@34 732 : forall P : unit -> Type, P tt -> forall u : unit, P u
adamc@34 733 ]]
adamc@34 734
adamc@34 735 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@34 736
adamc@34 737 To prove that [unit_rect] is nothing special, we can reimplement it manually. *)
adamc@34 738
adamc@34 739 Definition unit_rect' (P : unit -> Type) (f : P tt) (u : unit) :=
adamc@34 740 match u return (P u) with
adamc@34 741 | tt => f
adamc@34 742 end.
adamc@34 743
adamc@34 744 (** 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@34 745
adamc@34 746 We can check the implement of [nat_rect] as well: *)
adamc@34 747
adamc@34 748 Print nat_rect.
adamc@34 749 (** [[
adamc@34 750
adamc@34 751 nat_rect =
adamc@34 752 fun (P : nat -> Type) (f : P O) (f0 : forall n : nat, P n -> P (S n)) =>
adamc@34 753 fix F (n : nat) : P n :=
adamc@34 754 match n as n0 return (P n0) with
adamc@34 755 | O => f
adamc@34 756 | S n0 => f0 n0 (F n0)
adamc@34 757 end
adamc@34 758 : forall P : nat -> Type,
adamc@34 759 P O -> (forall n : nat, P n -> P (S n)) -> forall n : nat, P n
adamc@34 760 ]]
adamc@34 761
adamc@34 762 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@34 763
adamc@34 764 Section nat_ind'.
adamc@34 765 (** First, we have the property of natural numbers that we aim to prove. *)
adamc@34 766 Variable P : nat -> Prop.
adamc@34 767
adamc@34 768 (** Then we require a proof of the [O] case. *)
adamc@38 769 Hypothesis O_case : P O.
adamc@34 770
adamc@34 771 (** Next is a proof of the [S] case, which may assume an inductive hypothesis. *)
adamc@38 772 Hypothesis S_case : forall n : nat, P n -> P (S n).
adamc@34 773
adamc@34 774 (** Finally, we define a recursive function to tie the pieces together. *)
adamc@34 775 Fixpoint nat_ind' (n : nat) : P n :=
adamc@34 776 match n return (P n) with
adamc@34 777 | O => O_case
adamc@34 778 | S n' => S_case (nat_ind' n')
adamc@34 779 end.
adamc@34 780 End nat_ind'.
adamc@34 781
adamc@38 782 (** 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@34 783
adamc@34 784 %\medskip%
adamc@34 785
adamc@34 786 We can also examine the definition of [even_list_mut], which we generated with [Scheme] for a mutually-recursive type. *)
adamc@34 787
adamc@34 788 Print even_list_mut.
adamc@34 789 (** [[
adamc@34 790
adamc@34 791 even_list_mut =
adamc@34 792 fun (P : even_list -> Prop) (P0 : odd_list -> Prop)
adamc@34 793 (f : P ENil) (f0 : forall (n : nat) (o : odd_list), P0 o -> P (ECons n o))
adamc@34 794 (f1 : forall (n : nat) (e : even_list), P e -> P0 (OCons n e)) =>
adamc@34 795 fix F (e : even_list) : P e :=
adamc@34 796 match e as e0 return (P e0) with
adamc@34 797 | ENil => f
adamc@34 798 | ECons n o => f0 n o (F0 o)
adamc@34 799 end
adamc@34 800 with F0 (o : odd_list) : P0 o :=
adamc@34 801 match o as o0 return (P0 o0) with
adamc@34 802 | OCons n e => f1 n e (F e)
adamc@34 803 end
adamc@34 804 for F
adamc@34 805 : forall (P : even_list -> Prop) (P0 : odd_list -> Prop),
adamc@34 806 P ENil ->
adamc@34 807 (forall (n : nat) (o : odd_list), P0 o -> P (ECons n o)) ->
adamc@34 808 (forall (n : nat) (e : even_list), P e -> P0 (OCons n e)) ->
adamc@34 809 forall e : even_list, P e
adamc@34 810 ]]
adamc@34 811
adamc@34 812 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@34 813
adamc@34 814 Section even_list_mut'.
adamc@34 815 (** First, we need the properties that we are proving. *)
adamc@34 816 Variable Peven : even_list -> Prop.
adamc@34 817 Variable Podd : odd_list -> Prop.
adamc@34 818
adamc@34 819 (** Next, we need proofs of the three cases. *)
adamc@38 820 Hypothesis ENil_case : Peven ENil.
adamc@38 821 Hypothesis ECons_case : forall (n : nat) (o : odd_list), Podd o -> Peven (ECons n o).
adamc@38 822 Hypothesis OCons_case : forall (n : nat) (e : even_list), Peven e -> Podd (OCons n e).
adamc@34 823
adamc@34 824 (** Finally, we define the recursive functions. *)
adamc@34 825 Fixpoint even_list_mut' (e : even_list) : Peven e :=
adamc@34 826 match e return (Peven e) with
adamc@34 827 | ENil => ENil_case
adamc@34 828 | ECons n o => ECons_case n (odd_list_mut' o)
adamc@34 829 end
adamc@34 830 with odd_list_mut' (o : odd_list) : Podd o :=
adamc@34 831 match o return (Podd o) with
adamc@34 832 | OCons n e => OCons_case n (even_list_mut' e)
adamc@34 833 end.
adamc@34 834 End even_list_mut'.
adamc@34 835
adamc@34 836 (** 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@34 837
adamc@34 838 Section formula_ind'.
adamc@34 839 Variable P : formula -> Prop.
adamc@38 840 Hypothesis Eq_case : forall n1 n2 : nat, P (Eq n1 n2).
adamc@38 841 Hypothesis And_case : forall f1 f2 : formula,
adamc@34 842 P f1 -> P f2 -> P (And f1 f2).
adamc@38 843 Hypothesis Forall_case : forall f : nat -> formula,
adamc@34 844 (forall n : nat, P (f n)) -> P (Forall f).
adamc@34 845
adamc@34 846 Fixpoint formula_ind' (f : formula) : P f :=
adamc@34 847 match f return (P f) with
adamc@34 848 | Eq n1 n2 => Eq_case n1 n2
adamc@34 849 | And f1 f2 => And_case (formula_ind' f1) (formula_ind' f2)
adamc@34 850 | Forall f' => Forall_case f' (fun n => formula_ind' (f' n))
adamc@34 851 end.
adamc@34 852 End formula_ind'.
adamc@34 853
adamc@35 854
adamc@35 855 (** * Nested Inductive Types *)
adamc@35 856
adamc@35 857 (** 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@35 858
adamc@35 859 Inductive nat_tree : Set :=
adamc@35 860 | NLeaf' : nat_tree
adamc@35 861 | NNode' : nat -> list nat_tree -> nat_tree.
adamc@35 862
adamc@35 863 (** 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@35 864
adamc@35 865 Like we encountered for mutual inductive types, we find that the automatically-generated induction principle for [nat_tree] is too weak. *)
adamc@35 866
adamc@35 867 Check nat_tree_ind.
adamc@35 868 (** [[
adamc@35 869
adamc@35 870 nat_tree_ind
adamc@35 871 : forall P : nat_tree -> Prop,
adamc@35 872 P NLeaf' ->
adamc@35 873 (forall (n : nat) (l : list nat_tree), P (NNode' n l)) ->
adamc@35 874 forall n : nat_tree, P n
adamc@35 875 ]]
adamc@35 876
adamc@35 877 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@35 878
adamc@35 879 First, we will need an auxiliary definition, characterizing what it means for a property to hold of every element of a list. *)
adamc@35 880
adamc@35 881 Section All.
adamc@35 882 Variable T : Set.
adamc@35 883 Variable P : T -> Prop.
adamc@35 884
adamc@35 885 Fixpoint All (ls : list T) : Prop :=
adamc@35 886 match ls with
adamc@35 887 | Nil => True
adamc@35 888 | Cons h t => P h /\ All t
adamc@35 889 end.
adamc@35 890 End All.
adamc@35 891
adamc@35 892 (** It will be useful to look at the definitions of [True] and [/\], since we will want to write manual proofs of them below. *)
adamc@35 893
adamc@35 894 Print True.
adamc@35 895 (** [[
adamc@35 896
adamc@35 897 Inductive True : Prop := I : True
adamc@35 898 ]]
adamc@35 899
adamc@35 900 That is, [True] is a proposition with exactly one proof, [I], which we may always supply trivially.
adamc@35 901
adamc@35 902 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@35 903
adamc@35 904 Locate "/\".
adamc@35 905 (** [[
adamc@35 906
adamc@35 907 Notation Scope
adamc@35 908 "A /\ B" := and A B : type_scope
adamc@35 909 (default interpretation)
adamc@35 910 ]] *)
adamc@35 911
adamc@35 912 Print and.
adamc@35 913 (** [[
adamc@35 914
adamc@35 915 Inductive and (A : Prop) (B : Prop) : Prop := conj : A -> B -> A /\ B
adamc@35 916 For conj: Arguments A, B are implicit
adamc@35 917 For and: Argument scopes are [type_scope type_scope]
adamc@35 918 For conj: Argument scopes are [type_scope type_scope _ _]
adamc@35 919 ]]
adamc@35 920
adamc@35 921 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@35 922
adamc@35 923 %\medskip%
adamc@35 924
adamc@35 925 Now we create a section for our induction principle, following the same basic plan as in the last section of this chapter. *)
adamc@35 926
adamc@35 927 Section nat_tree_ind'.
adamc@35 928 Variable P : nat_tree -> Prop.
adamc@35 929
adamc@38 930 Hypothesis NLeaf'_case : P NLeaf'.
adamc@38 931 Hypothesis NNode'_case : forall (n : nat) (ls : list nat_tree),
adamc@35 932 All P ls -> P (NNode' n ls).
adamc@35 933
adamc@35 934 (** A first attempt at writing the induction principle itself follows the intuition that nested inductive type definitions are expanded into mutual inductive definitions.
adamc@35 935
adamc@35 936 [[
adamc@35 937
adamc@35 938 Fixpoint nat_tree_ind' (tr : nat_tree) : P tr :=
adamc@35 939 match tr return (P tr) with
adamc@35 940 | NLeaf' => NLeaf'_case
adamc@35 941 | NNode' n ls => NNode'_case n ls (list_nat_tree_ind ls)
adamc@35 942 end
adamc@35 943
adamc@35 944 with list_nat_tree_ind (ls : list nat_tree) : All P ls :=
adamc@35 945 match ls return (All P ls) with
adamc@35 946 | Nil => I
adamc@35 947 | Cons tr rest => conj (nat_tree_ind' tr) (list_nat_tree_ind rest)
adamc@35 948 end.
adamc@35 949
adamc@35 950 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@35 951
adamc@35 952 Fixpoint nat_tree_ind' (tr : nat_tree) : P tr :=
adamc@35 953 match tr return (P tr) with
adamc@35 954 | NLeaf' => NLeaf'_case
adamc@35 955 | NNode' n ls => NNode'_case n ls
adamc@35 956 ((fix list_nat_tree_ind (ls : list nat_tree) : All P ls :=
adamc@35 957 match ls return (All P ls) with
adamc@35 958 | Nil => I
adamc@35 959 | Cons tr rest => conj (nat_tree_ind' tr) (list_nat_tree_ind rest)
adamc@35 960 end) ls)
adamc@35 961 end.
adamc@35 962
adamc@35 963 (** 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@35 964
adamc@35 965 End nat_tree_ind'.
adamc@35 966
adamc@35 967 (** 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@35 968
adamc@35 969 Section map.
adamc@35 970 Variables T T' : Set.
adamc@35 971 Variable f : T -> T'.
adamc@35 972
adamc@35 973 Fixpoint map (ls : list T) : list T' :=
adamc@35 974 match ls with
adamc@35 975 | Nil => Nil
adamc@35 976 | Cons h t => Cons (f h) (map t)
adamc@35 977 end.
adamc@35 978 End map.
adamc@35 979
adamc@35 980 Fixpoint sum (ls : list nat) : nat :=
adamc@35 981 match ls with
adamc@35 982 | Nil => O
adamc@35 983 | Cons h t => plus h (sum t)
adamc@35 984 end.
adamc@35 985
adamc@35 986 (** Now we can define a size function over our trees. *)
adamc@35 987
adamc@35 988 Fixpoint ntsize (tr : nat_tree) : nat :=
adamc@35 989 match tr with
adamc@35 990 | NLeaf' => S O
adamc@35 991 | NNode' _ trs => S (sum (map ntsize trs))
adamc@35 992 end.
adamc@35 993
adamc@35 994 (** 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@35 995
adamc@35 996 Fixpoint ntsplice (tr1 tr2 : nat_tree) {struct tr1} : nat_tree :=
adamc@35 997 match tr1 with
adamc@35 998 | NLeaf' => NNode' O (Cons tr2 Nil)
adamc@35 999 | NNode' n Nil => NNode' n (Cons tr2 Nil)
adamc@35 1000 | NNode' n (Cons tr trs) => NNode' n (Cons (ntsplice tr tr2) trs)
adamc@35 1001 end.
adamc@35 1002
adamc@35 1003 (** 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 1004
adamc@41 1005 (* begin thide *)
adamc@35 1006 Lemma plus_S : forall n1 n2 : nat,
adamc@35 1007 plus n1 (S n2) = S (plus n1 n2).
adamc@35 1008 induction n1; crush.
adamc@35 1009 Qed.
adamc@41 1010 (* end thide *)
adamc@35 1011
adamc@35 1012 (** Now we begin the proof of the theorem, adding the lemma [plus_S] as a hint. *)
adamc@35 1013
adamc@35 1014 Theorem ntsize_ntsplice : forall tr1 tr2 : nat_tree, ntsize (ntsplice tr1 tr2)
adamc@35 1015 = plus (ntsize tr2) (ntsize tr1).
adamc@41 1016 (* begin thide *)
adamc@35 1017 Hint Rewrite plus_S : cpdt.
adamc@35 1018
adamc@35 1019 (** 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 1020 induction tr1 using nat_tree_ind'; crush.
adamc@35 1021
adamc@35 1022 (** One subgoal remains: [[
adamc@35 1023
adamc@35 1024 n : nat
adamc@35 1025 ls : list nat_tree
adamc@35 1026 H : All
adamc@35 1027 (fun tr1 : nat_tree =>
adamc@35 1028 forall tr2 : nat_tree,
adamc@35 1029 ntsize (ntsplice tr1 tr2) = plus (ntsize tr2) (ntsize tr1)) ls
adamc@35 1030 tr2 : nat_tree
adamc@35 1031 ============================
adamc@35 1032 ntsize
adamc@35 1033 match ls with
adamc@35 1034 | Nil => NNode' n (Cons tr2 Nil)
adamc@35 1035 | Cons tr trs => NNode' n (Cons (ntsplice tr tr2) trs)
adamc@35 1036 end = S (plus (ntsize tr2) (sum (map ntsize ls)))
adamc@35 1037 ]]
adamc@35 1038
adamc@35 1039 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 1040
adamc@35 1041 destruct ls; crush.
adamc@35 1042
adamc@36 1043 (** We can go further in automating the proof by exploiting the hint mechanism. *)
adamc@35 1044
adamc@35 1045 Restart.
adamc@35 1046 Hint Extern 1 (ntsize (match ?LS with Nil => _ | Cons _ _ => _ end) = _) =>
adamc@35 1047 destruct LS; crush.
adamc@35 1048 induction tr1 using nat_tree_ind'; crush.
adamc@35 1049 Qed.
adamc@41 1050 (* end thide *)
adamc@35 1051
adamc@35 1052 (** 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 1053
adamc@40 1054 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 1055
adamc@36 1056
adamc@36 1057 (** * Manual Proofs About Constructors *)
adamc@36 1058
adamc@36 1059 (** 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 1060
adamc@36 1061 Theorem true_neq_false : true <> false.
adamc@41 1062 (* begin thide *)
adamc@36 1063 (** We begin with the tactic [red], which is short for "one step of reduction," to unfold the definition of logical negation. *)
adamc@36 1064
adamc@36 1065 red.
adamc@36 1066 (** [[
adamc@36 1067
adamc@36 1068 ============================
adamc@36 1069 true = false -> False
adamc@36 1070 ]]
adamc@36 1071
adamc@36 1072 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 1073
adamc@36 1074 intro H.
adamc@36 1075 (** [[
adamc@36 1076
adamc@36 1077 H : true = false
adamc@36 1078 ============================
adamc@36 1079 False
adamc@36 1080 ]]
adamc@36 1081
adamc@36 1082 This is the point in the proof where we apply some creativity. We define a function whose utility will become clear soon. *)
adamc@36 1083
adamc@36 1084 Definition f (b : bool) := if b then True else False.
adamc@36 1085
adamc@36 1086 (** 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 1087
adamc@36 1088 change (f false).
adamc@36 1089 (** [[
adamc@36 1090
adamc@36 1091 H : true = false
adamc@36 1092 ============================
adamc@36 1093 f false
adamc@36 1094 ]]
adamc@36 1095
adamc@36 1096 Now the righthand side of [H]'s equality appears in the conclusion, so we can rewrite. *)
adamc@36 1097
adamc@36 1098 rewrite <- H.
adamc@36 1099 (** [[
adamc@36 1100
adamc@36 1101 H : true = false
adamc@36 1102 ============================
adamc@36 1103 f true
adamc@36 1104 ]]
adamc@36 1105
adamc@36 1106 We are almost done. Just how close we are to done is revealed by computational simplification. *)
adamc@36 1107
adamc@36 1108 simpl.
adamc@36 1109 (** [[
adamc@36 1110
adamc@36 1111 H : true = false
adamc@36 1112 ============================
adamc@36 1113 True
adamc@36 1114 ]] *)
adamc@36 1115
adamc@36 1116 trivial.
adamc@36 1117 Qed.
adamc@41 1118 (* end thide *)
adamc@36 1119
adamc@36 1120 (** I have no trivial automated version of this proof to suggest, beyond using [discriminate] or [congruence] in the first place.
adamc@36 1121
adamc@36 1122 %\medskip%
adamc@36 1123
adamc@36 1124 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 1125
adamc@36 1126 Theorem S_inj' : forall n m : nat, S n = S m -> n = m.
adamc@41 1127 (* begin thide *)
adamc@36 1128 intros n m H.
adamc@36 1129 change (pred (S n) = pred (S m)).
adamc@36 1130 rewrite H.
adamc@36 1131 reflexivity.
adamc@36 1132 Qed.
adamc@41 1133 (* end thide *)
adamc@36 1134
adamc@37 1135
adamc@37 1136 (** * Exercises *)
adamc@37 1137
adamc@37 1138 (** %\begin{enumerate}%#<ol>#
adamc@37 1139
adamc@37 1140 %\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 1141
adamc@39 1142 %\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 1143
adamc@39 1144 %\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 1145
adamc@38 1146 %\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 1147
adamc@38 1148 %\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 1149
adamc@38 1150 %\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 1151
adamc@37 1152 #</ol>#%\end{enumerate}% *)