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PC's Chapter 5 comments
author Adam Chlipala <adam@chlipala.net>
date Fri, 15 Oct 2010 09:50:34 -0400
parents fabbc71abd80
children 2c88fc1dbe33
rev   line source
adam@281 1 (* Copyright (c) 2008-2010, 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
adam@279 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. Most of our examples reproduce functionality from the Coq standard library, and we have tried to copy the standard library's choices of identifiers, where possible, so many of the definitions here are already available in the default Coq environment. *)
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@208 38 (** [unit : Set] *)
adamc@26 39
adamc@26 40 Check tt.
adamc@208 41 (** [tt : unit] *)
adamc@26 42
adamc@26 43 (** We can prove that [unit] is a genuine singleton type. *)
adamc@26 44
adamc@26 45 Theorem unit_singleton : forall x : unit, x = tt.
adamc@208 46
adamc@26 47 (** 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@208 48
adamc@41 49 (* begin thide *)
adamc@26 50 induction x.
adamc@26 51
adamc@208 52 (** The goal changes to:
adamc@208 53 [[
adamc@26 54 tt = tt
adamc@26 55 ]] *)
adamc@208 56
adamc@26 57 (** ...which we can discharge trivially. *)
adamc@208 58
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@205 66
adamc@205 67 ]]
adamc@205 68
adamc@208 69 %\noindent%...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 70
adamc@26 71 What exactly %\textit{%#<i>#is#</i>#%}% the induction principle for [unit]? We can ask Coq: *)
adamc@26 72
adamc@26 73 Check unit_ind.
adamc@208 74 (** [unit_ind : forall P : unit -> Prop, P tt -> forall u : unit, P u] *)
adamc@26 75
adamc@208 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@208 101 (** [Empty_set_ind : forall (P : Empty_set -> Prop) (e : Empty_set), P e] *)
adamc@26 102
adamc@208 103 (** 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@26 104
adamc@26 105 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@26 106
adamc@26 107 Definition e2u (e : Empty_set) : unit := match e with end.
adamc@26 108
adamc@26 109 (** 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@26 110
adamc@26 111 %\medskip%
adamc@26 112
adamc@26 113 Moving up the ladder of complexity, we can define the booleans: *)
adamc@26 114
adamc@26 115 Inductive bool : Set :=
adamc@26 116 | true
adamc@26 117 | false.
adamc@26 118
adamc@26 119 (** We can use less vacuous pattern matching to define boolean negation. *)
adamc@26 120
adam@279 121 Definition negb (b : bool) : bool :=
adamc@26 122 match b with
adamc@26 123 | true => false
adamc@26 124 | false => true
adamc@26 125 end.
adamc@26 126
adamc@27 127 (** An alternative definition desugars to the above: *)
adamc@27 128
adam@279 129 Definition negb' (b : bool) : bool :=
adamc@27 130 if b then false else true.
adamc@27 131
adam@279 132 (** We might want to prove that [negb] is its own inverse operation. *)
adamc@26 133
adam@279 134 Theorem negb_inverse : forall b : bool, negb (negb b) = b.
adamc@41 135 (* begin thide *)
adamc@26 136 destruct b.
adamc@26 137
adamc@208 138 (** After we case-analyze on [b], we are left with one subgoal for each constructor of [bool].
adamc@26 139
adamc@26 140 [[
adamc@26 141 2 subgoals
adamc@26 142
adamc@26 143 ============================
adam@279 144 negb (negb true) = true
adamc@26 145 ]]
adamc@26 146
adamc@26 147 [[
adamc@26 148 subgoal 2 is:
adam@279 149 negb (negb false) = false
adamc@208 150
adamc@26 151 ]]
adamc@26 152
adamc@26 153 The first subgoal follows by Coq's rules of computation, so we can dispatch it easily: *)
adamc@26 154
adamc@26 155 reflexivity.
adamc@26 156
adamc@26 157 (** Likewise for the second subgoal, so we can restart the proof and give a very compact justification. *)
adamc@26 158
adamc@26 159 Restart.
adamc@26 160 destruct b; reflexivity.
adamc@26 161 Qed.
adamc@41 162 (* end thide *)
adamc@27 163
adamc@27 164 (** Another theorem about booleans illustrates another useful tactic. *)
adamc@27 165
adam@279 166 Theorem negb_ineq : forall b : bool, negb b <> b.
adamc@41 167 (* begin thide *)
adamc@27 168 destruct b; discriminate.
adamc@27 169 Qed.
adamc@41 170 (* end thide *)
adamc@27 171
adamc@27 172 (** [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@27 173
adamc@27 174 At this point, it is probably not hard to guess what the underlying induction principle for [bool] is. *)
adamc@27 175
adamc@27 176 Check bool_ind.
adamc@208 177 (** [bool_ind : forall P : bool -> Prop, P true -> P false -> forall b : bool, P b] *)
adamc@28 178
adamc@28 179
adamc@28 180 (** * Simple Recursive Types *)
adamc@28 181
adamc@28 182 (** The natural numbers are the simplest common example of an inductive type that actually deserves the name. *)
adamc@28 183
adamc@28 184 Inductive nat : Set :=
adamc@28 185 | O : nat
adamc@28 186 | S : nat -> nat.
adamc@28 187
adamc@28 188 (** [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@28 189
adamc@28 190 Pattern matching works as we demonstrated in the last chapter: *)
adamc@28 191
adamc@28 192 Definition isZero (n : nat) : bool :=
adamc@28 193 match n with
adamc@28 194 | O => true
adamc@28 195 | S _ => false
adamc@28 196 end.
adamc@28 197
adamc@28 198 Definition pred (n : nat) : nat :=
adamc@28 199 match n with
adamc@28 200 | O => O
adamc@28 201 | S n' => n'
adamc@28 202 end.
adamc@28 203
adamc@28 204 (** We can prove theorems by case analysis: *)
adamc@28 205
adamc@28 206 Theorem S_isZero : forall n : nat, isZero (pred (S (S n))) = false.
adamc@41 207 (* begin thide *)
adamc@28 208 destruct n; reflexivity.
adamc@28 209 Qed.
adamc@41 210 (* end thide *)
adamc@28 211
adamc@28 212 (** We can also now get into genuine inductive theorems. First, we will need a recursive function, to make things interesting. *)
adamc@28 213
adamc@208 214 Fixpoint plus (n m : nat) : nat :=
adamc@28 215 match n with
adamc@28 216 | O => m
adamc@28 217 | S n' => S (plus n' m)
adamc@28 218 end.
adamc@28 219
adamc@208 220 (** Recall that [Fixpoint] is Coq's mechanism for recursive function definitions. Some theorems about [plus] can be proved without induction. *)
adamc@28 221
adamc@28 222 Theorem O_plus_n : forall n : nat, plus O n = n.
adamc@41 223 (* begin thide *)
adamc@28 224 intro; reflexivity.
adamc@28 225 Qed.
adamc@41 226 (* end thide *)
adamc@28 227
adamc@208 228 (** Coq's computation rules automatically simplify the application of [plus], because unfolding the definition of [plus] gives us a [match] expression where the branch to be taken is obvious from syntax alone. If we just reverse the order of the arguments, though, this no longer works, and we need induction. *)
adamc@28 229
adamc@28 230 Theorem n_plus_O : forall n : nat, plus n O = n.
adamc@41 231 (* begin thide *)
adamc@28 232 induction n.
adamc@28 233
adamc@28 234 (** Our first subgoal is [plus O O = O], which %\textit{%#<i>#is#</i>#%}% trivial by computation. *)
adamc@28 235
adamc@28 236 reflexivity.
adamc@28 237
adamc@28 238 (** Our second subgoal is more work and also demonstrates our first inductive hypothesis.
adamc@28 239
adamc@28 240 [[
adamc@28 241 n : nat
adamc@28 242 IHn : plus n O = n
adamc@28 243 ============================
adamc@28 244 plus (S n) O = S n
adamc@208 245
adamc@28 246 ]]
adamc@28 247
adamc@28 248 We can start out by using computation to simplify the goal as far as we can. *)
adamc@28 249
adamc@28 250 simpl.
adamc@28 251
adamc@28 252 (** Now the conclusion is [S (plus n O) = S n]. Using our inductive hypothesis: *)
adamc@28 253
adamc@28 254 rewrite IHn.
adamc@28 255
adamc@28 256 (** ...we get a trivial conclusion [S n = S n]. *)
adamc@28 257
adamc@28 258 reflexivity.
adamc@28 259
adamc@28 260 (** Not much really went on in this proof, so the [crush] tactic from the [Tactics] module can prove this theorem automatically. *)
adamc@28 261
adamc@28 262 Restart.
adamc@28 263 induction n; crush.
adamc@28 264 Qed.
adamc@41 265 (* end thide *)
adamc@28 266
adamc@28 267 (** We can check out the induction principle at work here: *)
adamc@28 268
adamc@28 269 Check nat_ind.
adamc@208 270 (** %\vspace{-.15in}% [[
adamc@208 271 nat_ind : forall P : nat -> Prop,
adamc@208 272 P O -> (forall n : nat, P n -> P (S n)) -> forall n : nat, P n
adamc@208 273
adamc@208 274 ]]
adamc@28 275
adamc@208 276 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@28 277
adamc@28 278 Since [nat] has a constructor that takes an argument, we may sometimes need to know that that constructor is injective. *)
adamc@28 279
adamc@28 280 Theorem S_inj : forall n m : nat, S n = S m -> n = m.
adamc@41 281 (* begin thide *)
adamc@28 282 injection 1; trivial.
adamc@28 283 Qed.
adamc@41 284 (* end thide *)
adamc@28 285
adamc@28 286 (** [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@28 287
adamc@29 288 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@29 289
adamc@29 290 %\medskip%
adamc@29 291
adamc@29 292 We can define a type of lists of natural numbers. *)
adamc@29 293
adamc@29 294 Inductive nat_list : Set :=
adamc@29 295 | NNil : nat_list
adamc@29 296 | NCons : nat -> nat_list -> nat_list.
adamc@29 297
adamc@29 298 (** Recursive definitions are straightforward extensions of what we have seen before. *)
adamc@29 299
adamc@29 300 Fixpoint nlength (ls : nat_list) : nat :=
adamc@29 301 match ls with
adamc@29 302 | NNil => O
adamc@29 303 | NCons _ ls' => S (nlength ls')
adamc@29 304 end.
adamc@29 305
adamc@208 306 Fixpoint napp (ls1 ls2 : nat_list) : nat_list :=
adamc@29 307 match ls1 with
adamc@29 308 | NNil => ls2
adamc@29 309 | NCons n ls1' => NCons n (napp ls1' ls2)
adamc@29 310 end.
adamc@29 311
adamc@29 312 (** Inductive theorem proving can again be automated quite effectively. *)
adamc@29 313
adamc@29 314 Theorem nlength_napp : forall ls1 ls2 : nat_list, nlength (napp ls1 ls2)
adamc@29 315 = plus (nlength ls1) (nlength ls2).
adamc@41 316 (* begin thide *)
adamc@29 317 induction ls1; crush.
adamc@29 318 Qed.
adamc@41 319 (* end thide *)
adamc@29 320
adamc@29 321 Check nat_list_ind.
adamc@208 322 (** %\vspace{-.15in}% [[
adamc@208 323 nat_list_ind
adamc@29 324 : forall P : nat_list -> Prop,
adamc@29 325 P NNil ->
adamc@29 326 (forall (n : nat) (n0 : nat_list), P n0 -> P (NCons n n0)) ->
adamc@29 327 forall n : nat_list, P n
adamc@29 328 ]]
adamc@29 329
adamc@29 330 %\medskip%
adamc@29 331
adamc@29 332 In general, we can implement any "tree" types as inductive types. For example, here are binary trees of naturals. *)
adamc@29 333
adamc@29 334 Inductive nat_btree : Set :=
adamc@29 335 | NLeaf : nat_btree
adamc@29 336 | NNode : nat_btree -> nat -> nat_btree -> nat_btree.
adamc@29 337
adamc@29 338 Fixpoint nsize (tr : nat_btree) : nat :=
adamc@29 339 match tr with
adamc@35 340 | NLeaf => S O
adamc@29 341 | NNode tr1 _ tr2 => plus (nsize tr1) (nsize tr2)
adamc@29 342 end.
adamc@29 343
adamc@208 344 Fixpoint nsplice (tr1 tr2 : nat_btree) : nat_btree :=
adamc@29 345 match tr1 with
adamc@35 346 | NLeaf => NNode tr2 O NLeaf
adamc@29 347 | NNode tr1' n tr2' => NNode (nsplice tr1' tr2) n tr2'
adamc@29 348 end.
adamc@29 349
adamc@29 350 Theorem plus_assoc : forall n1 n2 n3 : nat, plus (plus n1 n2) n3 = plus n1 (plus n2 n3).
adamc@41 351 (* begin thide *)
adamc@29 352 induction n1; crush.
adamc@29 353 Qed.
adamc@41 354 (* end thide *)
adamc@29 355
adamc@29 356 Theorem nsize_nsplice : forall tr1 tr2 : nat_btree, nsize (nsplice tr1 tr2)
adamc@29 357 = plus (nsize tr2) (nsize tr1).
adamc@41 358 (* begin thide *)
adamc@29 359 Hint Rewrite n_plus_O plus_assoc : cpdt.
adamc@29 360
adamc@29 361 induction tr1; crush.
adamc@29 362 Qed.
adamc@41 363 (* end thide *)
adamc@29 364
adamc@29 365 Check nat_btree_ind.
adamc@208 366 (** %\vspace{-.15in}% [[
adamc@208 367 nat_btree_ind
adamc@29 368 : forall P : nat_btree -> Prop,
adamc@29 369 P NLeaf ->
adamc@29 370 (forall n : nat_btree,
adamc@29 371 P n -> forall (n0 : nat) (n1 : nat_btree), P n1 -> P (NNode n n0 n1)) ->
adamc@29 372 forall n : nat_btree, P n
adamc@29 373 ]] *)
adamc@30 374
adamc@30 375
adamc@30 376 (** * Parameterized Types *)
adamc@30 377
adamc@30 378 (** We can also define polymorphic inductive types, as with algebraic datatypes in Haskell and ML. *)
adamc@30 379
adamc@30 380 Inductive list (T : Set) : Set :=
adamc@30 381 | Nil : list T
adamc@30 382 | Cons : T -> list T -> list T.
adamc@30 383
adamc@30 384 Fixpoint length T (ls : list T) : nat :=
adamc@30 385 match ls with
adamc@30 386 | Nil => O
adamc@30 387 | Cons _ ls' => S (length ls')
adamc@30 388 end.
adamc@30 389
adamc@208 390 Fixpoint app T (ls1 ls2 : list T) : list T :=
adamc@30 391 match ls1 with
adamc@30 392 | Nil => ls2
adamc@30 393 | Cons x ls1' => Cons x (app ls1' ls2)
adamc@30 394 end.
adamc@30 395
adamc@30 396 Theorem length_app : forall T (ls1 ls2 : list T), length (app ls1 ls2)
adamc@30 397 = plus (length ls1) (length ls2).
adamc@41 398 (* begin thide *)
adamc@30 399 induction ls1; crush.
adamc@30 400 Qed.
adamc@41 401 (* end thide *)
adamc@30 402
adamc@30 403 (** 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@30 404
adamc@30 405 (* begin hide *)
adamc@30 406 Reset list.
adamc@30 407 (* end hide *)
adamc@30 408
adamc@30 409 Section list.
adamc@30 410 Variable T : Set.
adamc@30 411
adamc@30 412 Inductive list : Set :=
adamc@30 413 | Nil : list
adamc@30 414 | Cons : T -> list -> list.
adamc@30 415
adamc@30 416 Fixpoint length (ls : list) : nat :=
adamc@30 417 match ls with
adamc@30 418 | Nil => O
adamc@30 419 | Cons _ ls' => S (length ls')
adamc@30 420 end.
adamc@30 421
adamc@208 422 Fixpoint app (ls1 ls2 : list) : list :=
adamc@30 423 match ls1 with
adamc@30 424 | Nil => ls2
adamc@30 425 | Cons x ls1' => Cons x (app ls1' ls2)
adamc@30 426 end.
adamc@30 427
adamc@30 428 Theorem length_app : forall ls1 ls2 : list, length (app ls1 ls2)
adamc@30 429 = plus (length ls1) (length ls2).
adamc@41 430 (* begin thide *)
adamc@30 431 induction ls1; crush.
adamc@30 432 Qed.
adamc@41 433 (* end thide *)
adamc@30 434 End list.
adamc@30 435
adamc@35 436 (* begin hide *)
adamc@35 437 Implicit Arguments Nil [T].
adamc@35 438 (* end hide *)
adamc@35 439
adamc@210 440 (** After we end the section, the [Variable]s we used are added as extra function parameters for each defined identifier, as needed. We verify that this has happened using the [Print] command, a cousin of [Check] which shows the definition of a symbol, rather than just its type. *)
adamc@30 441
adamc@202 442 Print list.
adamc@208 443 (** %\vspace{-.15in}% [[
adamc@208 444 Inductive list (T : Set) : Set :=
adamc@202 445 Nil : list T | Cons : T -> list T -> list Tlist
adamc@208 446
adamc@202 447 ]]
adamc@30 448
adamc@202 449 The final definition is the same as what we wrote manually before. The other elements of the section are altered similarly, turning out exactly as they were before, though we managed to write their definitions more succinctly. *)
adamc@30 450
adamc@30 451 Check length.
adamc@208 452 (** %\vspace{-.15in}% [[
adamc@208 453 length
adamc@30 454 : forall T : Set, list T -> nat
adamc@30 455 ]]
adamc@30 456
adamc@202 457 The parameter [T] is treated as a new argument to the induction principle, too. *)
adamc@30 458
adamc@30 459 Check list_ind.
adamc@208 460 (** %\vspace{-.15in}% [[
adamc@208 461 list_ind
adamc@30 462 : forall (T : Set) (P : list T -> Prop),
adamc@30 463 P (Nil T) ->
adamc@30 464 (forall (t : T) (l : list T), P l -> P (Cons t l)) ->
adamc@30 465 forall l : list T, P l
adamc@30 466 ]]
adamc@30 467
adamc@30 468 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@31 469
adamc@31 470
adamc@31 471 (** * Mutually Inductive Types *)
adamc@31 472
adamc@31 473 (** We can define inductive types that refer to each other: *)
adamc@31 474
adamc@31 475 Inductive even_list : Set :=
adamc@31 476 | ENil : even_list
adamc@31 477 | ECons : nat -> odd_list -> even_list
adamc@31 478
adamc@31 479 with odd_list : Set :=
adamc@31 480 | OCons : nat -> even_list -> odd_list.
adamc@31 481
adamc@31 482 Fixpoint elength (el : even_list) : nat :=
adamc@31 483 match el with
adamc@31 484 | ENil => O
adamc@31 485 | ECons _ ol => S (olength ol)
adamc@31 486 end
adamc@31 487
adamc@31 488 with olength (ol : odd_list) : nat :=
adamc@31 489 match ol with
adamc@31 490 | OCons _ el => S (elength el)
adamc@31 491 end.
adamc@31 492
adamc@208 493 Fixpoint eapp (el1 el2 : even_list) : even_list :=
adamc@31 494 match el1 with
adamc@31 495 | ENil => el2
adamc@31 496 | ECons n ol => ECons n (oapp ol el2)
adamc@31 497 end
adamc@31 498
adamc@208 499 with oapp (ol : odd_list) (el : even_list) : odd_list :=
adamc@31 500 match ol with
adamc@31 501 | OCons n el' => OCons n (eapp el' el)
adamc@31 502 end.
adamc@31 503
adamc@31 504 (** Everything is going roughly the same as in past examples, until we try to prove a theorem similar to those that came before. *)
adamc@31 505
adamc@31 506 Theorem elength_eapp : forall el1 el2 : even_list,
adamc@31 507 elength (eapp el1 el2) = plus (elength el1) (elength el2).
adamc@41 508 (* begin thide *)
adamc@31 509 induction el1; crush.
adamc@31 510
adamc@31 511 (** One goal remains: [[
adamc@31 512
adamc@31 513 n : nat
adamc@31 514 o : odd_list
adamc@31 515 el2 : even_list
adamc@31 516 ============================
adamc@31 517 S (olength (oapp o el2)) = S (plus (olength o) (elength el2))
adamc@31 518 ]]
adamc@31 519
adamc@31 520 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@31 521
adamc@31 522 Abort.
adamc@31 523 Check even_list_ind.
adamc@208 524 (** %\vspace{-.15in}% [[
adamc@208 525 even_list_ind
adamc@31 526 : forall P : even_list -> Prop,
adamc@31 527 P ENil ->
adamc@31 528 (forall (n : nat) (o : odd_list), P (ECons n o)) ->
adamc@31 529 forall e : even_list, P e
adamc@208 530
adamc@31 531 ]]
adamc@31 532
adamc@31 533 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@31 534
adamc@31 535 Scheme even_list_mut := Induction for even_list Sort Prop
adamc@31 536 with odd_list_mut := Induction for odd_list Sort Prop.
adamc@31 537
adamc@31 538 Check even_list_mut.
adamc@208 539 (** %\vspace{-.15in}% [[
adamc@208 540 even_list_mut
adamc@31 541 : forall (P : even_list -> Prop) (P0 : odd_list -> Prop),
adamc@31 542 P ENil ->
adamc@31 543 (forall (n : nat) (o : odd_list), P0 o -> P (ECons n o)) ->
adamc@31 544 (forall (n : nat) (e : even_list), P e -> P0 (OCons n e)) ->
adamc@31 545 forall e : even_list, P e
adamc@208 546
adamc@31 547 ]]
adamc@31 548
adamc@31 549 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@31 550
adamc@31 551 Theorem n_plus_O' : forall n : nat, plus n O = n.
adamc@31 552 apply (nat_ind (fun n => plus n O = n)); crush.
adamc@31 553 Qed.
adamc@31 554
adamc@31 555 (** 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@31 556
adamc@31 557 This technique generalizes to our mutual example: *)
adamc@31 558
adamc@31 559 Theorem elength_eapp : forall el1 el2 : even_list,
adamc@31 560 elength (eapp el1 el2) = plus (elength el1) (elength el2).
adamc@41 561
adamc@31 562 apply (even_list_mut
adamc@31 563 (fun el1 : even_list => forall el2 : even_list,
adamc@31 564 elength (eapp el1 el2) = plus (elength el1) (elength el2))
adamc@31 565 (fun ol : odd_list => forall el : even_list,
adamc@31 566 olength (oapp ol el) = plus (olength ol) (elength el))); crush.
adamc@31 567 Qed.
adamc@41 568 (* end thide *)
adamc@31 569
adamc@31 570 (** 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@33 571
adamc@33 572
adamc@33 573 (** * Reflexive Types *)
adamc@33 574
adamc@33 575 (** 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@33 576
adamc@33 577 Inductive formula : Set :=
adamc@33 578 | Eq : nat -> nat -> formula
adamc@33 579 | And : formula -> formula -> formula
adamc@33 580 | Forall : (nat -> formula) -> formula.
adamc@33 581
adamc@33 582 (** 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@33 583
adamc@33 584 Example forall_refl : formula := Forall (fun x => Eq x x).
adamc@33 585
adamc@33 586 (** We can write recursive functions over reflexive types quite naturally. Here is one translating our formulas into native Coq propositions. *)
adamc@33 587
adamc@33 588 Fixpoint formulaDenote (f : formula) : Prop :=
adamc@33 589 match f with
adamc@33 590 | Eq n1 n2 => n1 = n2
adamc@33 591 | And f1 f2 => formulaDenote f1 /\ formulaDenote f2
adamc@33 592 | Forall f' => forall n : nat, formulaDenote (f' n)
adamc@33 593 end.
adamc@33 594
adamc@33 595 (** We can also encode a trivial formula transformation that swaps the order of equality and conjunction operands. *)
adamc@33 596
adamc@33 597 Fixpoint swapper (f : formula) : formula :=
adamc@33 598 match f with
adamc@33 599 | Eq n1 n2 => Eq n2 n1
adamc@33 600 | And f1 f2 => And (swapper f2) (swapper f1)
adamc@33 601 | Forall f' => Forall (fun n => swapper (f' n))
adamc@33 602 end.
adamc@33 603
adamc@33 604 (** It is helpful to prove that this transformation does not make true formulas false. *)
adamc@33 605
adamc@33 606 Theorem swapper_preserves_truth : forall f, formulaDenote f -> formulaDenote (swapper f).
adamc@41 607 (* begin thide *)
adamc@33 608 induction f; crush.
adamc@33 609 Qed.
adamc@41 610 (* end thide *)
adamc@33 611
adamc@33 612 (** We can take a look at the induction principle behind this proof. *)
adamc@33 613
adamc@33 614 Check formula_ind.
adamc@208 615 (** %\vspace{-.15in}% [[
adamc@208 616 formula_ind
adamc@33 617 : forall P : formula -> Prop,
adamc@33 618 (forall n n0 : nat, P (Eq n n0)) ->
adamc@33 619 (forall f0 : formula,
adamc@33 620 P f0 -> forall f1 : formula, P f1 -> P (And f0 f1)) ->
adamc@33 621 (forall f1 : nat -> formula,
adamc@33 622 (forall n : nat, P (f1 n)) -> P (Forall f1)) ->
adamc@33 623 forall f2 : formula, P f2
adamc@208 624
adamc@208 625 ]]
adamc@33 626
adamc@208 627 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@33 628
adamc@33 629 %\medskip%
adamc@33 630
adamc@33 631 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@33 632
adamc@33 633 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@33 634
adamc@33 635 (** [[
adamc@33 636 Inductive term : Set :=
adamc@33 637 | App : term -> term -> term
adamc@33 638 | Abs : (term -> term) -> term.
adamc@33 639
adamc@33 640 Error: Non strictly positive occurrence of "term" in "(term -> term) -> term"
adamc@208 641
adamc@33 642 ]]
adamc@33 643
adamc@33 644 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@33 645
adamc@33 646 Why must Coq enforce this restriction? Imagine that our last definition had been accepted, allowing us to write this function:
adamc@33 647
adamc@33 648 [[
adamc@33 649 Definition uhoh (t : term) : term :=
adamc@33 650 match t with
adamc@33 651 | Abs f => f t
adamc@33 652 | _ => t
adamc@33 653 end.
adamc@33 654
adamc@205 655 ]]
adamc@205 656
adamc@33 657 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@33 658
adamc@33 659 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@33 660
adamc@33 661 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@34 662
adamc@34 663
adamc@34 664 (** * An Interlude on Proof Terms *)
adamc@34 665
adamc@34 666 (** 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@34 667
adamc@34 668 Print unit_ind.
adamc@208 669 (** %\vspace{-.15in}% [[
adamc@208 670 unit_ind =
adamc@208 671 fun P : unit -> Prop => unit_rect P
adamc@34 672 : forall P : unit -> Prop, P tt -> forall u : unit, P u
adamc@208 673
adamc@34 674 ]]
adamc@34 675
adamc@34 676 We see that this induction principle is defined in terms of a more general principle, [unit_rect]. *)
adamc@34 677
adamc@34 678 Check unit_rect.
adamc@208 679 (** %\vspace{-.15in}% [[
adamc@208 680 unit_rect
adamc@34 681 : forall P : unit -> Type, P tt -> forall u : unit, P u
adamc@208 682
adamc@34 683 ]]
adamc@34 684
adamc@34 685 [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@34 686
adamc@34 687 Print unit_rec.
adamc@208 688 (** %\vspace{-.15in}% [[
adamc@208 689 unit_rec =
adamc@208 690 fun P : unit -> Set => unit_rect P
adamc@34 691 : forall P : unit -> Set, P tt -> forall u : unit, P u
adamc@208 692
adamc@34 693 ]]
adamc@34 694
adamc@34 695 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@34 696
adamc@34 697 Definition always_O (u : unit) : nat :=
adamc@34 698 match u with
adamc@34 699 | tt => O
adamc@34 700 end.
adamc@34 701
adamc@34 702 Definition always_O' (u : unit) : nat :=
adamc@34 703 unit_rec (fun _ : unit => nat) O u.
adamc@34 704
adamc@34 705 (** 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@34 706
adamc@34 707 Print unit_rect.
adamc@208 708 (** %\vspace{-.15in}% [[
adamc@208 709 unit_rect =
adamc@208 710 fun (P : unit -> Type) (f : P tt) (u : unit) =>
adamc@208 711 match u as u0 return (P u0) with
adamc@208 712 | tt => f
adamc@208 713 end
adamc@34 714 : forall P : unit -> Type, P tt -> forall u : unit, P u
adamc@208 715
adamc@34 716 ]]
adamc@34 717
adamc@34 718 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@34 719
adamc@34 720 To prove that [unit_rect] is nothing special, we can reimplement it manually. *)
adamc@34 721
adamc@34 722 Definition unit_rect' (P : unit -> Type) (f : P tt) (u : unit) :=
adamc@208 723 match u with
adamc@34 724 | tt => f
adamc@34 725 end.
adamc@34 726
adamc@208 727 (** We rely on Coq's heuristics for inferring [match] annotations.
adamc@34 728
adamc@208 729 We can check the implementation of [nat_rect] as well: *)
adamc@34 730
adamc@34 731 Print nat_rect.
adamc@208 732 (** %\vspace{-.15in}% [[
adamc@208 733 nat_rect =
adamc@208 734 fun (P : nat -> Type) (f : P O) (f0 : forall n : nat, P n -> P (S n)) =>
adamc@208 735 fix F (n : nat) : P n :=
adamc@208 736 match n as n0 return (P n0) with
adamc@208 737 | O => f
adamc@208 738 | S n0 => f0 n0 (F n0)
adamc@208 739 end
adamc@208 740 : forall P : nat -> Type,
adamc@208 741 P O -> (forall n : nat, P n -> P (S n)) -> forall n : nat, P n
adamc@208 742 ]]
adamc@34 743
adamc@208 744 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@34 745
adamc@208 746 Section nat_ind'.
adamc@208 747 (** First, we have the property of natural numbers that we aim to prove. *)
adamc@34 748
adamc@208 749 Variable P : nat -> Prop.
adamc@34 750
adamc@208 751 (** Then we require a proof of the [O] case. *)
adamc@34 752
adamc@208 753 Hypothesis O_case : P O.
adamc@34 754
adamc@208 755 (** Next is a proof of the [S] case, which may assume an inductive hypothesis. *)
adamc@34 756
adamc@208 757 Hypothesis S_case : forall n : nat, P n -> P (S n).
adamc@34 758
adamc@208 759 (** Finally, we define a recursive function to tie the pieces together. *)
adamc@34 760
adamc@208 761 Fixpoint nat_ind' (n : nat) : P n :=
adamc@208 762 match n with
adamc@208 763 | O => O_case
adamc@208 764 | S n' => S_case (nat_ind' n')
adamc@208 765 end.
adamc@208 766 End nat_ind'.
adamc@34 767
adamc@208 768 (** 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@34 769
adamc@208 770 %\medskip%
adamc@34 771
adamc@208 772 We can also examine the definition of [even_list_mut], which we generated with [Scheme] for a mutually-recursive type. *)
adamc@34 773
adamc@208 774 Print even_list_mut.
adamc@208 775 (** %\vspace{-.15in}% [[
adamc@208 776 even_list_mut =
adamc@208 777 fun (P : even_list -> Prop) (P0 : odd_list -> Prop)
adamc@208 778 (f : P ENil) (f0 : forall (n : nat) (o : odd_list), P0 o -> P (ECons n o))
adamc@208 779 (f1 : forall (n : nat) (e : even_list), P e -> P0 (OCons n e)) =>
adamc@208 780 fix F (e : even_list) : P e :=
adamc@208 781 match e as e0 return (P e0) with
adamc@208 782 | ENil => f
adamc@208 783 | ECons n o => f0 n o (F0 o)
adamc@208 784 end
adamc@208 785 with F0 (o : odd_list) : P0 o :=
adamc@208 786 match o as o0 return (P0 o0) with
adamc@208 787 | OCons n e => f1 n e (F e)
adamc@208 788 end
adamc@208 789 for F
adamc@208 790 : forall (P : even_list -> Prop) (P0 : odd_list -> Prop),
adamc@208 791 P ENil ->
adamc@208 792 (forall (n : nat) (o : odd_list), P0 o -> P (ECons n o)) ->
adamc@208 793 (forall (n : nat) (e : even_list), P e -> P0 (OCons n e)) ->
adamc@208 794 forall e : even_list, P e
adamc@34 795
adamc@208 796 ]]
adamc@34 797
adamc@208 798 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@208 799
adamc@208 800 Section even_list_mut'.
adamc@208 801 (** First, we need the properties that we are proving. *)
adamc@208 802
adamc@208 803 Variable Peven : even_list -> Prop.
adamc@208 804 Variable Podd : odd_list -> Prop.
adamc@208 805
adamc@208 806 (** Next, we need proofs of the three cases. *)
adamc@208 807
adamc@208 808 Hypothesis ENil_case : Peven ENil.
adamc@208 809 Hypothesis ECons_case : forall (n : nat) (o : odd_list), Podd o -> Peven (ECons n o).
adamc@208 810 Hypothesis OCons_case : forall (n : nat) (e : even_list), Peven e -> Podd (OCons n e).
adamc@208 811
adamc@208 812 (** Finally, we define the recursive functions. *)
adamc@208 813
adamc@208 814 Fixpoint even_list_mut' (e : even_list) : Peven e :=
adamc@208 815 match e with
adamc@208 816 | ENil => ENil_case
adamc@208 817 | ECons n o => ECons_case n (odd_list_mut' o)
adamc@208 818 end
adamc@208 819 with odd_list_mut' (o : odd_list) : Podd o :=
adamc@208 820 match o with
adamc@208 821 | OCons n e => OCons_case n (even_list_mut' e)
adamc@208 822 end.
adamc@34 823 End even_list_mut'.
adamc@34 824
adamc@34 825 (** 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@34 826
adamc@34 827 Section formula_ind'.
adamc@34 828 Variable P : formula -> Prop.
adamc@38 829 Hypothesis Eq_case : forall n1 n2 : nat, P (Eq n1 n2).
adamc@38 830 Hypothesis And_case : forall f1 f2 : formula,
adamc@34 831 P f1 -> P f2 -> P (And f1 f2).
adamc@38 832 Hypothesis Forall_case : forall f : nat -> formula,
adamc@34 833 (forall n : nat, P (f n)) -> P (Forall f).
adamc@34 834
adamc@34 835 Fixpoint formula_ind' (f : formula) : P f :=
adamc@208 836 match f with
adamc@34 837 | Eq n1 n2 => Eq_case n1 n2
adamc@34 838 | And f1 f2 => And_case (formula_ind' f1) (formula_ind' f2)
adamc@34 839 | Forall f' => Forall_case f' (fun n => formula_ind' (f' n))
adamc@34 840 end.
adamc@34 841 End formula_ind'.
adamc@34 842
adamc@35 843
adamc@35 844 (** * Nested Inductive Types *)
adamc@35 845
adamc@35 846 (** 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@35 847
adamc@35 848 Inductive nat_tree : Set :=
adamc@35 849 | NLeaf' : nat_tree
adamc@35 850 | NNode' : nat -> list nat_tree -> nat_tree.
adamc@35 851
adamc@35 852 (** 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@35 853
adamc@35 854 Like we encountered for mutual inductive types, we find that the automatically-generated induction principle for [nat_tree] is too weak. *)
adamc@35 855
adamc@35 856 Check nat_tree_ind.
adamc@208 857 (** %\vspace{-.15in}% [[
adamc@208 858 nat_tree_ind
adamc@35 859 : forall P : nat_tree -> Prop,
adamc@35 860 P NLeaf' ->
adamc@35 861 (forall (n : nat) (l : list nat_tree), P (NNode' n l)) ->
adamc@35 862 forall n : nat_tree, P n
adamc@208 863
adamc@35 864 ]]
adamc@35 865
adamc@35 866 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@35 867
adamc@35 868 First, we will need an auxiliary definition, characterizing what it means for a property to hold of every element of a list. *)
adamc@35 869
adamc@35 870 Section All.
adamc@35 871 Variable T : Set.
adamc@35 872 Variable P : T -> Prop.
adamc@35 873
adamc@35 874 Fixpoint All (ls : list T) : Prop :=
adamc@35 875 match ls with
adamc@35 876 | Nil => True
adamc@35 877 | Cons h t => P h /\ All t
adamc@35 878 end.
adamc@35 879 End All.
adamc@35 880
adamc@35 881 (** It will be useful to look at the definitions of [True] and [/\], since we will want to write manual proofs of them below. *)
adamc@35 882
adamc@35 883 Print True.
adamc@208 884 (** %\vspace{-.15in}% [[
adamc@208 885 Inductive True : Prop := I : True
adamc@208 886
adamc@208 887 ]]
adamc@35 888
adamc@35 889 That is, [True] is a proposition with exactly one proof, [I], which we may always supply trivially.
adamc@35 890
adamc@35 891 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@35 892
adamc@35 893 Locate "/\".
adamc@208 894 (** %\vspace{-.15in}% [[
adamc@208 895 Notation Scope
adamc@208 896 "A /\ B" := and A B : type_scope
adamc@208 897 (default interpretation)
adamc@35 898 ]] *)
adamc@35 899
adamc@35 900 Print and.
adamc@208 901 (** %\vspace{-.15in}% [[
adamc@208 902 Inductive and (A : Prop) (B : Prop) : Prop := conj : A -> B -> A /\ B
adamc@208 903 For conj: Arguments A, B are implicit
adamc@208 904 For and: Argument scopes are [type_scope type_scope]
adamc@208 905 For conj: Argument scopes are [type_scope type_scope _ _]
adamc@208 906
adamc@35 907 ]]
adamc@35 908
adamc@35 909 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@35 910
adamc@35 911 %\medskip%
adamc@35 912
adamc@35 913 Now we create a section for our induction principle, following the same basic plan as in the last section of this chapter. *)
adamc@35 914
adamc@35 915 Section nat_tree_ind'.
adamc@35 916 Variable P : nat_tree -> Prop.
adamc@35 917
adamc@38 918 Hypothesis NLeaf'_case : P NLeaf'.
adamc@38 919 Hypothesis NNode'_case : forall (n : nat) (ls : list nat_tree),
adamc@35 920 All P ls -> P (NNode' n ls).
adamc@35 921
adamc@35 922 (** A first attempt at writing the induction principle itself follows the intuition that nested inductive type definitions are expanded into mutual inductive definitions.
adamc@35 923
adamc@35 924 [[
adamc@35 925 Fixpoint nat_tree_ind' (tr : nat_tree) : P tr :=
adamc@208 926 match tr with
adamc@35 927 | NLeaf' => NLeaf'_case
adamc@35 928 | NNode' n ls => NNode'_case n ls (list_nat_tree_ind ls)
adamc@35 929 end
adamc@35 930
adamc@35 931 with list_nat_tree_ind (ls : list nat_tree) : All P ls :=
adamc@208 932 match ls with
adamc@35 933 | Nil => I
adamc@35 934 | Cons tr rest => conj (nat_tree_ind' tr) (list_nat_tree_ind rest)
adamc@35 935 end.
adamc@35 936
adamc@205 937 ]]
adamc@205 938
adam@279 939 Coq rejects this definition, saying "Recursive call to nat_tree_ind' has principal argument equal to "tr" instead of rest." There is no deep theoretical reason why this program should be rejected; Coq applies incomplete termination-checking heuristics, and it is necessary to learn a few of the most important rules. The term "nested inductive type" hints at the solution to this particular problem. Just like true mutually-inductive types require mutually-recursive induction principles, nested types require nested recursion. *)
adamc@35 940
adamc@35 941 Fixpoint nat_tree_ind' (tr : nat_tree) : P tr :=
adamc@208 942 match tr with
adamc@35 943 | NLeaf' => NLeaf'_case
adamc@35 944 | NNode' n ls => NNode'_case n ls
adamc@35 945 ((fix list_nat_tree_ind (ls : list nat_tree) : All P ls :=
adamc@208 946 match ls with
adamc@35 947 | Nil => I
adamc@35 948 | Cons tr rest => conj (nat_tree_ind' tr) (list_nat_tree_ind rest)
adamc@35 949 end) ls)
adamc@35 950 end.
adamc@35 951
adam@279 952 (** 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@35 953
adamc@35 954 End nat_tree_ind'.
adamc@35 955
adamc@35 956 (** 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@35 957
adamc@35 958 Section map.
adamc@35 959 Variables T T' : Set.
adamc@35 960 Variable f : T -> T'.
adamc@35 961
adamc@35 962 Fixpoint map (ls : list T) : list T' :=
adamc@35 963 match ls with
adamc@35 964 | Nil => Nil
adamc@35 965 | Cons h t => Cons (f h) (map t)
adamc@35 966 end.
adamc@35 967 End map.
adamc@35 968
adamc@35 969 Fixpoint sum (ls : list nat) : nat :=
adamc@35 970 match ls with
adamc@35 971 | Nil => O
adamc@35 972 | Cons h t => plus h (sum t)
adamc@35 973 end.
adamc@35 974
adamc@35 975 (** Now we can define a size function over our trees. *)
adamc@35 976
adamc@35 977 Fixpoint ntsize (tr : nat_tree) : nat :=
adamc@35 978 match tr with
adamc@35 979 | NLeaf' => S O
adamc@35 980 | NNode' _ trs => S (sum (map ntsize trs))
adamc@35 981 end.
adamc@35 982
adamc@35 983 (** 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@35 984
adamc@208 985 Fixpoint ntsplice (tr1 tr2 : nat_tree) : nat_tree :=
adamc@35 986 match tr1 with
adamc@35 987 | NLeaf' => NNode' O (Cons tr2 Nil)
adamc@35 988 | NNode' n Nil => NNode' n (Cons tr2 Nil)
adamc@35 989 | NNode' n (Cons tr trs) => NNode' n (Cons (ntsplice tr tr2) trs)
adamc@35 990 end.
adamc@35 991
adamc@35 992 (** 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@35 993
adamc@41 994 (* begin thide *)
adamc@35 995 Lemma plus_S : forall n1 n2 : nat,
adamc@35 996 plus n1 (S n2) = S (plus n1 n2).
adamc@35 997 induction n1; crush.
adamc@35 998 Qed.
adamc@41 999 (* end thide *)
adamc@35 1000
adamc@35 1001 (** Now we begin the proof of the theorem, adding the lemma [plus_S] as a hint. *)
adamc@35 1002
adamc@35 1003 Theorem ntsize_ntsplice : forall tr1 tr2 : nat_tree, ntsize (ntsplice tr1 tr2)
adamc@35 1004 = plus (ntsize tr2) (ntsize tr1).
adamc@41 1005 (* begin thide *)
adamc@35 1006 Hint Rewrite plus_S : cpdt.
adamc@35 1007
adamc@35 1008 (** 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@208 1009
adamc@35 1010 induction tr1 using nat_tree_ind'; crush.
adamc@35 1011
adamc@35 1012 (** One subgoal remains: [[
adamc@35 1013 n : nat
adamc@35 1014 ls : list nat_tree
adamc@35 1015 H : All
adamc@35 1016 (fun tr1 : nat_tree =>
adamc@35 1017 forall tr2 : nat_tree,
adamc@35 1018 ntsize (ntsplice tr1 tr2) = plus (ntsize tr2) (ntsize tr1)) ls
adamc@35 1019 tr2 : nat_tree
adamc@35 1020 ============================
adamc@35 1021 ntsize
adamc@35 1022 match ls with
adamc@35 1023 | Nil => NNode' n (Cons tr2 Nil)
adamc@35 1024 | Cons tr trs => NNode' n (Cons (ntsplice tr tr2) trs)
adamc@35 1025 end = S (plus (ntsize tr2) (sum (map ntsize ls)))
adamc@208 1026
adamc@35 1027 ]]
adamc@35 1028
adamc@35 1029 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 1030
adamc@35 1031 destruct ls; crush.
adamc@35 1032
adamc@36 1033 (** We can go further in automating the proof by exploiting the hint mechanism. *)
adamc@35 1034
adamc@35 1035 Restart.
adamc@35 1036 Hint Extern 1 (ntsize (match ?LS with Nil => _ | Cons _ _ => _ end) = _) =>
adamc@35 1037 destruct LS; crush.
adamc@35 1038 induction tr1 using nat_tree_ind'; crush.
adamc@35 1039 Qed.
adamc@41 1040 (* end thide *)
adamc@35 1041
adamc@35 1042 (** 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 1043
adamc@40 1044 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 1045
adamc@36 1046
adamc@36 1047 (** * Manual Proofs About Constructors *)
adamc@36 1048
adamc@36 1049 (** 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 1050
adamc@36 1051 Theorem true_neq_false : true <> false.
adamc@208 1052
adamc@41 1053 (* begin thide *)
adamc@36 1054 (** We begin with the tactic [red], which is short for "one step of reduction," to unfold the definition of logical negation. *)
adamc@36 1055
adamc@36 1056 red.
adamc@36 1057 (** [[
adamc@36 1058 ============================
adamc@36 1059 true = false -> False
adamc@208 1060
adamc@36 1061 ]]
adamc@36 1062
adamc@36 1063 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 1064
adamc@36 1065 intro H.
adamc@36 1066 (** [[
adamc@36 1067 H : true = false
adamc@36 1068 ============================
adamc@36 1069 False
adamc@208 1070
adamc@36 1071 ]]
adamc@36 1072
adamc@36 1073 This is the point in the proof where we apply some creativity. We define a function whose utility will become clear soon. *)
adamc@36 1074
adamc@36 1075 Definition f (b : bool) := if b then True else False.
adamc@36 1076
adamc@36 1077 (** 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 1078
adamc@36 1079 change (f false).
adamc@36 1080 (** [[
adamc@36 1081 H : true = false
adamc@36 1082 ============================
adamc@36 1083 f false
adamc@208 1084
adamc@36 1085 ]]
adamc@36 1086
adamc@202 1087 Now the righthand side of [H]'s equality appears in the conclusion, so we can rewrite, using the notation [<-] to request to replace the righthand side the equality with the lefthand side. *)
adamc@36 1088
adamc@36 1089 rewrite <- H.
adamc@36 1090 (** [[
adamc@36 1091 H : true = false
adamc@36 1092 ============================
adamc@36 1093 f true
adamc@208 1094
adamc@36 1095 ]]
adamc@36 1096
adamc@36 1097 We are almost done. Just how close we are to done is revealed by computational simplification. *)
adamc@36 1098
adamc@36 1099 simpl.
adamc@36 1100 (** [[
adamc@36 1101 H : true = false
adamc@36 1102 ============================
adamc@36 1103 True
adamc@208 1104
adamc@36 1105 ]] *)
adamc@36 1106
adamc@36 1107 trivial.
adamc@36 1108 Qed.
adamc@41 1109 (* end thide *)
adamc@36 1110
adamc@36 1111 (** I have no trivial automated version of this proof to suggest, beyond using [discriminate] or [congruence] in the first place.
adamc@36 1112
adamc@36 1113 %\medskip%
adamc@36 1114
adamc@36 1115 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 1116
adamc@36 1117 Theorem S_inj' : forall n m : nat, S n = S m -> n = m.
adamc@41 1118 (* begin thide *)
adamc@36 1119 intros n m H.
adamc@36 1120 change (pred (S n) = pred (S m)).
adamc@36 1121 rewrite H.
adamc@36 1122 reflexivity.
adamc@36 1123 Qed.
adamc@41 1124 (* end thide *)
adamc@36 1125
adamc@37 1126
adamc@37 1127 (** * Exercises *)
adamc@37 1128
adamc@37 1129 (** %\begin{enumerate}%#<ol>#
adamc@37 1130
adamc@201 1131 %\item%#<li># Define an inductive type [truth] with three constructors, [Yes], [No], and [Maybe]. [Yes] stands for certain truth, [No] 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 1132
adamc@39 1133 %\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 1134
adamc@39 1135 %\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 1136
adamc@38 1137 %\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 1138
adamc@38 1139 %\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 1140
adamc@38 1141 %\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 1142
adamc@37 1143 #</ol>#%\end{enumerate}% *)