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A pass of improvements to vertical spacing, up through end of InductiveTypes
author Adam Chlipala <adam@chlipala.net>
date Mon, 30 Jul 2012 13:21:36 -0400
parents 5f25705a10ea
children f923024bd284
rev   line source
adam@394 1 (* Copyright (c) 2008-2012, Adam Chlipala
adamc@45 2 *
adamc@45 3 * This work is licensed under a
adamc@45 4 * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
adamc@45 5 * Unported License.
adamc@45 6 * The license text is available at:
adamc@45 7 * http://creativecommons.org/licenses/by-nc-nd/3.0/
adamc@45 8 *)
adamc@45 9
adamc@45 10 (* begin hide *)
adamc@45 11 Require Import List.
adamc@45 12
adam@314 13 Require Import CpdtTactics.
adamc@45 14
adamc@45 15 Set Implicit Arguments.
adam@322 16
adam@322 17 (* Extra definitions to get coqdoc to choose the right fonts. *)
adam@322 18
adam@323 19 (* begin thide *)
adam@322 20 Inductive unit := tt.
adam@322 21 Inductive Empty_set := .
adam@322 22 Inductive bool := true | false.
adam@322 23 Inductive sum := .
adam@322 24 Inductive prod := .
adam@322 25 Inductive and := conj.
adam@322 26 Inductive or := or_introl | or_intror.
adam@322 27 Inductive ex := ex_intro.
adam@426 28 Inductive eq := eq_refl.
adam@322 29 Reset unit.
adam@323 30 (* end thide *)
adamc@45 31 (* end hide *)
adamc@45 32
adamc@45 33 (** %\chapter{Inductive Predicates}% *)
adamc@45 34
adam@322 35 (** The so-called %\index{Curry-Howard correspondence}``%#"#Curry-Howard correspondence#"#%''~\cite{Curry,Howard}% states a formal connection between functional programs and mathematical proofs. In the last chapter, we snuck in a first introduction to this subject in Coq. Witness the close similarity between the types [unit] and [True] from the standard library: *)
adamc@45 36
adamc@45 37 Print unit.
adam@322 38 (** [[
adamc@209 39 Inductive unit : Set := tt : unit
adam@302 40 ]]
adam@302 41 *)
adamc@45 42
adamc@45 43 Print True.
adam@322 44 (** [[
adamc@209 45 Inductive True : Prop := I : True
adam@322 46 ]]
adam@302 47 *)
adamc@45 48
adam@350 49 (** Recall that [unit] is the type with only one value, and [True] is the proposition that always holds. Despite this superficial difference between the two concepts, in both cases we can use the same inductive definition mechanism. The connection goes further than this. We see that we arrive at the definition of [True] by replacing [unit] by [True], [tt] by [I], and [Set] by [Prop]. The first two of these differences are superficial changes of names, while the third difference is the crucial one for separating programs from proofs. A term [T] of type [Set] is a type of programs, and a term of type [T] is a program. A term [T] of type [Prop] is a logical proposition, and its proofs are of type [T]. Chapter 12 goes into more detail about the theoretical differences between [Prop] and [Set]. For now, we will simply follow common intuitions about what a proof is.
adamc@45 50
adam@398 51 The type [unit] has one value, [tt]. The type [True] has one proof, [I]. Why distinguish between these two types? Many people who have read about Curry-Howard in an abstract context and not put it to use in proof engineering answer that the two types in fact _should not_ be distinguished. There is a certain aesthetic appeal to this point of view, but I want to argue that it is best to treat Curry-Howard very loosely in practical proving. There are Coq-specific reasons for preferring the distinction, involving efficient compilation and avoidance of paradoxes in the presence of classical math, but I will argue that there is a more general principle that should lead us to avoid conflating programming and proving.
adamc@45 52
adam@401 53 The essence of the argument is roughly this: to an engineer, not all functions of type [A -> B] are created equal, but all proofs of a proposition [P -> Q] are. This idea is known as%\index{proof irrelevance}% _proof irrelevance_, and its formalizations in logics prevent us from distinguishing between alternate proofs of the same proposition. Proof irrelevance is compatible with, but not derivable in, Gallina. Apart from this theoretical concern, I will argue that it is most effective to do engineering with Coq by employing different techniques for programs versus proofs. Most of this book is organized around that distinction, describing how to program, by applying standard functional programming techniques in the presence of dependent types; and how to prove, by writing custom Ltac decision procedures.
adamc@45 54
adam@421 55 With that perspective in mind, this chapter is sort of a mirror image of the last chapter, introducing how to define predicates with inductive definitions. We will point out similarities in places, but much of the effective Coq user's bag of tricks is disjoint for predicates versus "datatypes." This chapter is also a covert introduction to dependent types, which are the foundation on which interesting inductive predicates are built, though we will rely on tactics to build dependently typed proof terms for us for now. A future chapter introduces more manual application of dependent types. *)
adamc@45 56
adamc@45 57
adamc@48 58 (** * Propositional Logic *)
adamc@45 59
adamc@45 60 (** Let us begin with a brief tour through the definitions of the connectives for propositional logic. We will work within a Coq section that provides us with a set of propositional variables. In Coq parlance, these are just terms of type [Prop.] *)
adamc@45 61
adamc@45 62 Section Propositional.
adamc@46 63 Variables P Q R : Prop.
adamc@45 64
adamc@45 65 (** In Coq, the most basic propositional connective is implication, written [->], which we have already used in almost every proof. Rather than being defined inductively, implication is built into Coq as the function type constructor.
adamc@45 66
adamc@45 67 We have also already seen the definition of [True]. For a demonstration of a lower-level way of establishing proofs of inductive predicates, we turn to this trivial theorem. *)
adamc@45 68
adamc@45 69 Theorem obvious : True.
adamc@55 70 (* begin thide *)
adamc@45 71 apply I.
adamc@55 72 (* end thide *)
adamc@45 73 Qed.
adamc@45 74
adam@401 75 (** We may always use the [apply] tactic to take a proof step based on applying a particular constructor of the inductive predicate that we are trying to establish. Sometimes there is only one constructor that could possibly apply, in which case a shortcut is available:%\index{tactics!constructor}% *)
adamc@45 76
adamc@55 77 (* begin thide *)
adamc@45 78 Theorem obvious' : True.
adamc@45 79 constructor.
adamc@45 80 Qed.
adamc@45 81
adamc@55 82 (* end thide *)
adamc@55 83
adamc@45 84 (** There is also a predicate [False], which is the Curry-Howard mirror image of [Empty_set] from the last chapter. *)
adamc@45 85
adamc@45 86 Print False.
adam@322 87 (** [[
adamc@209 88 Inductive False : Prop :=
adamc@209 89
adamc@209 90 ]]
adamc@45 91
adamc@209 92 We can conclude anything from [False], doing case analysis on a proof of [False] in the same way we might do case analysis on, say, a natural number. Since there are no cases to consider, any such case analysis succeeds immediately in proving the goal. *)
adamc@45 93
adamc@45 94 Theorem False_imp : False -> 2 + 2 = 5.
adamc@55 95 (* begin thide *)
adamc@45 96 destruct 1.
adamc@55 97 (* end thide *)
adamc@45 98 Qed.
adamc@45 99
adamc@45 100 (** In a consistent context, we can never build a proof of [False]. In inconsistent contexts that appear in the courses of proofs, it is usually easiest to proceed by demonstrating that inconsistency with an explicit proof of [False]. *)
adamc@45 101
adamc@45 102 Theorem arith_neq : 2 + 2 = 5 -> 9 + 9 = 835.
adamc@55 103 (* begin thide *)
adamc@45 104 intro.
adamc@45 105
adam@322 106 (** At this point, we have an inconsistent hypothesis [2 + 2 = 5], so the specific conclusion is not important. We use the %\index{tactics!elimtype}%[elimtype] tactic to state a proposition, telling Coq that we wish to construct a proof of the new proposition and then prove the original goal by case analysis on the structure of the new auxiliary proof. Since [False] has no constructors, [elimtype False] simply leaves us with the obligation to prove [False]. *)
adamc@45 107
adamc@45 108 elimtype False.
adamc@45 109 (** [[
adamc@45 110 H : 2 + 2 = 5
adamc@45 111 ============================
adamc@45 112 False
adamc@209 113
adamc@209 114 ]]
adamc@45 115
adamc@209 116 For now, we will leave the details of this proof about arithmetic to [crush]. *)
adamc@45 117
adamc@45 118 crush.
adamc@55 119 (* end thide *)
adamc@45 120 Qed.
adamc@45 121
adamc@45 122 (** A related notion to [False] is logical negation. *)
adamc@45 123
adam@421 124 (* begin hide *)
adam@421 125 Definition foo := not.
adam@421 126 (* end hide *)
adam@421 127
adamc@45 128 Print not.
adamc@209 129 (** %\vspace{-.15in}% [[
adamc@209 130 not = fun A : Prop => A -> False
adamc@209 131 : Prop -> Prop
adamc@209 132
adamc@209 133 ]]
adamc@45 134
adam@280 135 We see that [not] is just shorthand for implication of [False]. We can use that fact explicitly in proofs. The syntax [~ P] expands to [not P]. *)
adamc@45 136
adamc@45 137 Theorem arith_neq' : ~ (2 + 2 = 5).
adamc@55 138 (* begin thide *)
adamc@45 139 unfold not.
adamc@45 140 (** [[
adamc@45 141 ============================
adamc@45 142 2 + 2 = 5 -> False
adam@302 143 ]]
adam@302 144 *)
adamc@45 145
adamc@45 146 crush.
adamc@55 147 (* end thide *)
adamc@45 148 Qed.
adamc@45 149
adamc@45 150 (** We also have conjunction, which we introduced in the last chapter. *)
adamc@45 151
adamc@45 152 Print and.
adam@401 153 (** [[
adam@322 154 Inductive and (A : Prop) (B : Prop) : Prop := conj : A -> B -> A /\ B
adamc@209 155
adamc@209 156 ]]
adamc@209 157
adam@322 158 The interested reader can check that [and] has a Curry-Howard equivalent called %\index{Gallina terms!prod}%[prod], the type of pairs. However, it is generally most convenient to reason about conjunction using tactics. An explicit proof of commutativity of [and] illustrates the usual suspects for such tasks. The operator [/\] is an infix shorthand for [and]. *)
adamc@45 159
adamc@45 160 Theorem and_comm : P /\ Q -> Q /\ P.
adamc@209 161
adamc@55 162 (* begin thide *)
adamc@45 163 (** We start by case analysis on the proof of [P /\ Q]. *)
adamc@45 164
adamc@45 165 destruct 1.
adamc@45 166 (** [[
adamc@45 167 H : P
adamc@45 168 H0 : Q
adamc@45 169 ============================
adamc@45 170 Q /\ P
adamc@209 171
adamc@209 172 ]]
adamc@45 173
adam@322 174 Every proof of a conjunction provides proofs for both conjuncts, so we get a single subgoal reflecting that. We can proceed by splitting this subgoal into a case for each conjunct of [Q /\ P].%\index{tactics!split}% *)
adamc@45 175
adamc@45 176 split.
adam@439 177 (** 2 subgoals
adamc@45 178
adamc@45 179 H : P
adamc@45 180 H0 : Q
adamc@45 181 ============================
adamc@45 182 Q
adam@439 183
adam@439 184 subgoal 2 is
adam@439 185
adam@322 186 P
adamc@209 187
adamc@209 188 ]]
adamc@45 189
adam@322 190 In each case, the conclusion is among our hypotheses, so the %\index{tactics!assumption}%[assumption] tactic finishes the process. *)
adamc@45 191
adamc@45 192 assumption.
adamc@45 193 assumption.
adamc@55 194 (* end thide *)
adamc@45 195 Qed.
adamc@45 196
adam@322 197 (** Coq disjunction is called %\index{Gallina terms!or}%[or] and abbreviated with the infix operator [\/]. *)
adamc@45 198
adamc@45 199 Print or.
adam@401 200 (** [[
adamc@209 201 Inductive or (A : Prop) (B : Prop) : Prop :=
adamc@209 202 or_introl : A -> A \/ B | or_intror : B -> A \/ B
adamc@209 203
adamc@209 204 ]]
adamc@45 205
adam@322 206 We see that there are two ways to prove a disjunction: prove the first disjunct or prove the second. The Curry-Howard analogue of this is the Coq %\index{Gallina terms!sum}%[sum] type. We can demonstrate the main tactics here with another proof of commutativity. *)
adamc@45 207
adamc@45 208 Theorem or_comm : P \/ Q -> Q \/ P.
adamc@55 209
adamc@55 210 (* begin thide *)
adamc@45 211 (** As in the proof for [and], we begin with case analysis, though this time we are met by two cases instead of one. *)
adamc@209 212
adamc@45 213 destruct 1.
adam@439 214 (** [[
adam@439 215 2 subgoals
adamc@45 216
adamc@45 217 H : P
adamc@45 218 ============================
adamc@45 219 Q \/ P
adam@439 220
adam@439 221 subgoal 2 is
adam@439 222
adamc@45 223 Q \/ P
adamc@209 224
adamc@209 225 ]]
adamc@45 226
adam@401 227 We can see that, in the first subgoal, we want to prove the disjunction by proving its second disjunct. The %\index{tactics!right}%[right] tactic telegraphs this intent. *)
adam@322 228
adamc@45 229 right; assumption.
adamc@45 230
adam@322 231 (** The second subgoal has a symmetric proof.%\index{tactics!left}%
adamc@45 232
adamc@45 233 [[
adamc@45 234 1 subgoal
adamc@45 235
adamc@45 236 H : Q
adamc@45 237 ============================
adamc@45 238 Q \/ P
adam@302 239 ]]
adam@302 240 *)
adamc@45 241
adamc@45 242 left; assumption.
adam@322 243
adamc@55 244 (* end thide *)
adamc@45 245 Qed.
adamc@45 246
adamc@46 247
adamc@46 248 (* begin hide *)
adamc@46 249 (* In-class exercises *)
adamc@46 250
adamc@46 251 Theorem contra : P -> ~P -> R.
adamc@52 252 (* begin thide *)
adamc@52 253 unfold not.
adamc@52 254 intros.
adamc@52 255 elimtype False.
adamc@52 256 apply H0.
adamc@52 257 assumption.
adamc@52 258 (* end thide *)
adamc@46 259 Admitted.
adamc@46 260
adamc@46 261 Theorem and_assoc : (P /\ Q) /\ R -> P /\ (Q /\ R).
adamc@52 262 (* begin thide *)
adamc@52 263 intros.
adamc@52 264 destruct H.
adamc@52 265 destruct H.
adamc@52 266 split.
adamc@52 267 assumption.
adamc@52 268 split.
adamc@52 269 assumption.
adamc@52 270 assumption.
adamc@52 271 (* end thide *)
adamc@46 272 Admitted.
adamc@46 273
adamc@46 274 Theorem or_assoc : (P \/ Q) \/ R -> P \/ (Q \/ R).
adamc@52 275 (* begin thide *)
adamc@52 276 intros.
adamc@52 277 destruct H.
adamc@52 278 destruct H.
adamc@52 279 left.
adamc@52 280 assumption.
adamc@52 281 right.
adamc@52 282 left.
adamc@52 283 assumption.
adamc@52 284 right.
adamc@52 285 right.
adamc@52 286 assumption.
adamc@52 287 (* end thide *)
adamc@46 288 Admitted.
adamc@46 289
adamc@46 290 (* end hide *)
adamc@46 291
adamc@46 292
adam@421 293 (** It would be a shame to have to plod manually through all proofs about propositional logic. Luckily, there is no need. One of the most basic Coq automation tactics is %\index{tactics!tauto}%[tauto], which is a complete decision procedure for constructive propositional logic. (More on what "constructive" means in the next section.) We can use [tauto] to dispatch all of the purely propositional theorems we have proved so far. *)
adamc@46 294
adamc@46 295 Theorem or_comm' : P \/ Q -> Q \/ P.
adamc@55 296 (* begin thide *)
adamc@46 297 tauto.
adamc@55 298 (* end thide *)
adamc@46 299 Qed.
adamc@46 300
adam@401 301 (** Sometimes propositional reasoning forms important plumbing for the proof of a theorem, but we still need to apply some other smarts about, say, arithmetic. The tactic %\index{tactics!intuition}%[intuition] is a generalization of [tauto] that proves everything it can using propositional reasoning. When some goals remain, it uses propositional laws to simplify them as far as possible. Consider this example, which uses the list concatenation operator [++] from the standard library. *)
adamc@46 302
adamc@46 303 Theorem arith_comm : forall ls1 ls2 : list nat,
adamc@46 304 length ls1 = length ls2 \/ length ls1 + length ls2 = 6
adamc@46 305 -> length (ls1 ++ ls2) = 6 \/ length ls1 = length ls2.
adamc@55 306 (* begin thide *)
adamc@46 307 intuition.
adamc@46 308
adamc@46 309 (** A lot of the proof structure has been generated for us by [intuition], but the final proof depends on a fact about lists. The remaining subgoal hints at what cleverness we need to inject. *)
adamc@46 310
adamc@46 311 (** [[
adamc@46 312 ls1 : list nat
adamc@46 313 ls2 : list nat
adamc@46 314 H0 : length ls1 + length ls2 = 6
adamc@46 315 ============================
adamc@46 316 length (ls1 ++ ls2) = 6 \/ length ls1 = length ls2
adamc@209 317
adamc@209 318 ]]
adamc@46 319
adamc@209 320 We can see that we need a theorem about lengths of concatenated lists, which we proved last chapter and is also in the standard library. *)
adamc@46 321
adamc@46 322 rewrite app_length.
adamc@46 323 (** [[
adamc@46 324 ls1 : list nat
adamc@46 325 ls2 : list nat
adamc@46 326 H0 : length ls1 + length ls2 = 6
adamc@46 327 ============================
adamc@46 328 length ls1 + length ls2 = 6 \/ length ls1 = length ls2
adamc@209 329
adamc@209 330 ]]
adamc@46 331
adamc@209 332 Now the subgoal follows by purely propositional reasoning. That is, we could replace [length ls1 + length ls2 = 6] with [P] and [length ls1 = length ls2] with [Q] and arrive at a tautology of propositional logic. *)
adamc@46 333
adamc@46 334 tauto.
adamc@55 335 (* end thide *)
adamc@46 336 Qed.
adamc@46 337
adam@322 338 (** The [intuition] tactic is one of the main bits of glue in the implementation of [crush], so, with a little help, we can get a short automated proof of the theorem. *)
adamc@46 339
adamc@55 340 (* begin thide *)
adamc@46 341 Theorem arith_comm' : forall ls1 ls2 : list nat,
adamc@46 342 length ls1 = length ls2 \/ length ls1 + length ls2 = 6
adamc@46 343 -> length (ls1 ++ ls2) = 6 \/ length ls1 = length ls2.
adam@375 344 Hint Rewrite app_length.
adamc@46 345
adamc@46 346 crush.
adamc@46 347 Qed.
adamc@55 348 (* end thide *)
adamc@46 349
adamc@45 350 End Propositional.
adamc@45 351
adam@322 352 (** Ending the section here has the same effect as always. Each of our propositional theorems becomes universally quantified over the propositional variables that we used. *)
adam@322 353
adamc@46 354
adamc@47 355 (** * What Does It Mean to Be Constructive? *)
adamc@46 356
adam@401 357 (** One potential point of confusion in the presentation so far is the distinction between [bool] and [Prop]. The datatype [bool] is built from two values [true] and [false], while [Prop] is a more primitive type that includes among its members [True] and [False]. Why not collapse these two concepts into one, and why must there be more than two states of mathematical truth?
adamc@46 358
adam@421 359 The answer comes from the fact that Coq implements%\index{constructive logic}% _constructive_ or%\index{intuitionistic logic|see{constructive logic}}% _intuitionistic_ logic, in contrast to the%\index{classical logic}% _classical_ logic that you may be more familiar with. In constructive logic, classical tautologies like [~ ~ P -> P] and [P \/ ~ P] do not always hold. In general, we can only prove these tautologies when [P] is%\index{decidability}% _decidable_, in the sense of %\index{computability|see{decidability}}%computability theory. The Curry-Howard encoding that Coq uses for [or] allows us to extract either a proof of [P] or a proof of [~ P] from any proof of [P \/ ~ P]. Since our proofs are just functional programs which we can run, a general %\index{law of the excluded middle}%law of the excluded middle would give us a decision procedure for the halting problem, where the instantiations of [P] would be formulas like "this particular Turing machine halts."
adamc@47 360
adam@421 361 Hence the distinction between [bool] and [Prop]. Programs of type [bool] are computational by construction; we can always run them to determine their results. Many [Prop]s are undecidable, and so we can write more expressive formulas with [Prop]s than with [bool]s, but the inevitable consequence is that we cannot simply "run a [Prop] to determine its truth."
adamc@47 362
adam@401 363 Constructive logic lets us define all of the logical connectives in an aesthetically appealing way, with orthogonal inductive definitions. That is, each connective is defined independently using a simple, shared mechanism. Constructivity also enables a trick called%\index{program extraction}% _program extraction_, where we write programs by phrasing them as theorems to be proved. Since our proofs are just functional programs, we can extract executable programs from our final proofs, which we could not do as naturally with classical proofs.
adamc@47 364
adamc@47 365 We will see more about Coq's program extraction facility in a later chapter. However, I think it is worth interjecting another warning at this point, following up on the prior warning about taking the Curry-Howard correspondence too literally. It is possible to write programs by theorem-proving methods in Coq, but hardly anyone does it. It is almost always most useful to maintain the distinction between programs and proofs. If you write a program by proving a theorem, you are likely to run into algorithmic inefficiencies that you introduced in your proof to make it easier to prove. It is a shame to have to worry about such situations while proving tricky theorems, and it is a happy state of affairs that you almost certainly will not need to, with the ideal of extracting programs from proofs being confined mostly to theoretical studies. *)
adamc@48 366
adamc@48 367
adamc@48 368 (** * First-Order Logic *)
adamc@48 369
adam@421 370 (** The %\index{Gallina terms!forall}%[forall] connective of first-order logic, which we have seen in many examples so far, is built into Coq. Getting ahead of ourselves a bit, we can see it as the dependent function type constructor. In fact, implication and universal quantification are just different syntactic shorthands for the same Coq mechanism. A formula [P -> Q] is equivalent to [forall x : P, Q], where [x] does not appear in [Q]. That is, the "real" type of the implication says "for every proof of [P], there exists a proof of [Q]."
adamc@48 371
adam@322 372 %\index{existential quantification}\index{Gallina terms!exists}\index{Gallina terms!ex}%Existential quantification is defined in the standard library. *)
adamc@48 373
adam@322 374 Print ex.
adam@401 375 (** [[
adamc@209 376 Inductive ex (A : Type) (P : A -> Prop) : Prop :=
adamc@209 377 ex_intro : forall x : A, P x -> ex P
adamc@209 378
adamc@209 379 ]]
adamc@48 380
adam@322 381 The family [ex] is parameterized by the type [A] that we quantify over, and by a predicate [P] over [A]s. We prove an existential by exhibiting some [x] of type [A], along with a proof of [P x]. As usual, there are tactics that save us from worrying about the low-level details most of the time. We use the equality operator [=], which, depending on the settings in which they learned logic, different people will say either is or is not part of first-order logic. For our purposes, it is. *)
adamc@48 382
adamc@48 383 Theorem exist1 : exists x : nat, x + 1 = 2.
adamc@55 384 (* begin thide *)
adamc@67 385 (** remove printing exists *)
adam@421 386 (** We can start this proof with a tactic %\index{tactics!exists}%[exists], which should not be confused with the formula constructor shorthand of the same name. (In the PDF version of this document, the reverse %`%#'#E#'#%'% appears instead of the text "exists" in formulas.) *)
adamc@209 387
adamc@48 388 exists 1.
adamc@48 389
adamc@209 390 (** The conclusion is replaced with a version using the existential witness that we announced.
adamc@48 391
adamc@209 392 [[
adamc@48 393 ============================
adamc@48 394 1 + 1 = 2
adam@302 395 ]]
adam@302 396 *)
adamc@48 397
adamc@48 398 reflexivity.
adamc@55 399 (* end thide *)
adamc@48 400 Qed.
adamc@48 401
adamc@48 402 (** printing exists $\exists$ *)
adamc@48 403
adamc@48 404 (** We can also use tactics to reason about existential hypotheses. *)
adamc@48 405
adamc@48 406 Theorem exist2 : forall n m : nat, (exists x : nat, n + x = m) -> n <= m.
adamc@55 407 (* begin thide *)
adamc@48 408 (** We start by case analysis on the proof of the existential fact. *)
adamc@209 409
adamc@48 410 destruct 1.
adamc@48 411 (** [[
adamc@48 412 n : nat
adamc@48 413 m : nat
adamc@48 414 x : nat
adamc@48 415 H : n + x = m
adamc@48 416 ============================
adamc@48 417 n <= m
adamc@209 418
adamc@209 419 ]]
adamc@48 420
adamc@209 421 The goal has been replaced by a form where there is a new free variable [x], and where we have a new hypothesis that the body of the existential holds with [x] substituted for the old bound variable. From here, the proof is just about arithmetic and is easy to automate. *)
adamc@48 422
adamc@48 423 crush.
adamc@55 424 (* end thide *)
adamc@48 425 Qed.
adamc@48 426
adamc@48 427
adamc@48 428 (* begin hide *)
adamc@48 429 (* In-class exercises *)
adamc@48 430
adamc@48 431 Theorem forall_exists_commute : forall (A B : Type) (P : A -> B -> Prop),
adamc@48 432 (exists x : A, forall y : B, P x y) -> (forall y : B, exists x : A, P x y).
adamc@52 433 (* begin thide *)
adamc@52 434 intros.
adamc@52 435 destruct H.
adamc@52 436 exists x.
adamc@52 437 apply H.
adamc@52 438 (* end thide *)
adamc@48 439 Admitted.
adamc@48 440
adamc@48 441 (* end hide *)
adamc@48 442
adamc@48 443
adam@322 444 (** The tactic [intuition] has a first-order cousin called %\index{tactics!firstorder}%[firstorder], which proves many formulas when only first-order reasoning is needed, and it tries to perform first-order simplifications in any case. First-order reasoning is much harder than propositional reasoning, so [firstorder] is much more likely than [intuition] to get stuck in a way that makes it run for long enough to be useless. *)
adamc@49 445
adamc@49 446
adamc@49 447 (** * Predicates with Implicit Equality *)
adamc@49 448
adamc@49 449 (** We start our exploration of a more complicated class of predicates with a simple example: an alternative way of characterizing when a natural number is zero. *)
adamc@49 450
adamc@49 451 Inductive isZero : nat -> Prop :=
adamc@49 452 | IsZero : isZero 0.
adamc@49 453
adamc@49 454 Theorem isZero_zero : isZero 0.
adamc@55 455 (* begin thide *)
adamc@49 456 constructor.
adamc@55 457 (* end thide *)
adamc@49 458 Qed.
adamc@49 459
adam@401 460 (** We can call [isZero] a%\index{judgment}% _judgment_, in the sense often used in the semantics of programming languages. Judgments are typically defined in the style of%\index{natural deduction}% _natural deduction_, where we write a number of%\index{inference rules}% _inference rules_ with premises appearing above a solid line and a conclusion appearing below the line. In this example, the sole constructor [IsZero] of [isZero] can be thought of as the single inference rule for deducing [isZero], with nothing above the line and [isZero 0] below it. The proof of [isZero_zero] demonstrates how we can apply an inference rule.
adamc@49 461
adam@398 462 The definition of [isZero] differs in an important way from all of the other inductive definitions that we have seen in this and the previous chapter. Instead of writing just [Set] or [Prop] after the colon, here we write [nat -> Prop]. We saw examples of parameterized types like [list], but there the parameters appeared with names _before_ the colon. Every constructor of a parameterized inductive type must have a range type that uses the same parameter, whereas the form we use here enables us to use different arguments to the type for different constructors.
adamc@49 463
adam@322 464 For instance, our definition [isZero] makes the predicate provable only when the argument is [0]. We can see that the concept of equality is somehow implicit in the inductive definition mechanism. The way this is accomplished is similar to the way that logic variables are used in %\index{Prolog}%Prolog, and it is a very powerful mechanism that forms a foundation for formalizing all of mathematics. In fact, though it is natural to think of inductive types as folding in the functionality of equality, in Coq, the true situation is reversed, with equality defined as just another inductive type!%\index{Gallina terms!eq}\index{Gallina terms!refl\_equal}% *)
adamc@49 465
adamc@49 466 Print eq.
adam@401 467 (** [[
adam@426 468 Inductive eq (A : Type) (x : A) : A -> Prop := eq_refl : x = x
adamc@209 469
adamc@209 470 ]]
adamc@49 471
adam@426 472 Behind the scenes, uses of infix [=] are expanded to instances of [eq]. We see that [eq] has both a parameter [x] that is fixed and an extra unnamed argument of the same type. The type of [eq] allows us to state any equalities, even those that are provably false. However, examining the type of equality's sole constructor [eq_refl], we see that we can only _prove_ equality when its two arguments are syntactically equal. This definition turns out to capture all of the basic properties of equality, and the equality-manipulating tactics that we have seen so far, like [reflexivity] and [rewrite], are implemented treating [eq] as just another inductive type with a well-chosen definition. Another way of stating that definition is: equality is defined as the least reflexive relation.
adamc@49 473
adam@322 474 Returning to the example of [isZero], we can see how to work with hypotheses that use this predicate. *)
adamc@49 475
adamc@49 476 Theorem isZero_plus : forall n m : nat, isZero m -> n + m = n.
adamc@55 477 (* begin thide *)
adamc@49 478 (** We want to proceed by cases on the proof of the assumption about [isZero]. *)
adamc@209 479
adamc@49 480 destruct 1.
adamc@49 481 (** [[
adamc@49 482 n : nat
adamc@49 483 ============================
adamc@49 484 n + 0 = n
adamc@209 485
adamc@209 486 ]]
adamc@49 487
adamc@209 488 Since [isZero] has only one constructor, we are presented with only one subgoal. The argument [m] to [isZero] is replaced with that type's argument from the single constructor [IsZero]. From this point, the proof is trivial. *)
adamc@49 489
adamc@49 490 crush.
adamc@55 491 (* end thide *)
adamc@49 492 Qed.
adamc@49 493
adamc@49 494 (** Another example seems at first like it should admit an analogous proof, but in fact provides a demonstration of one of the most basic gotchas of Coq proving. *)
adamc@49 495
adamc@49 496 Theorem isZero_contra : isZero 1 -> False.
adamc@55 497 (* begin thide *)
adamc@49 498 (** Let us try a proof by cases on the assumption, as in the last proof. *)
adamc@209 499
adamc@49 500 destruct 1.
adamc@49 501 (** [[
adamc@49 502 ============================
adamc@49 503 False
adamc@209 504
adamc@209 505 ]]
adamc@49 506
adamc@209 507 It seems that case analysis has not helped us much at all! Our sole hypothesis disappears, leaving us, if anything, worse off than we were before. What went wrong? We have met an important restriction in tactics like [destruct] and [induction] when applied to types with arguments. If the arguments are not already free variables, they will be replaced by new free variables internally before doing the case analysis or induction. Since the argument [1] to [isZero] is replaced by a fresh variable, we lose the crucial fact that it is not equal to [0].
adamc@49 508
adam@421 509 Why does Coq use this restriction? We will discuss the issue in detail in a future chapter, when we see the dependently typed programming techniques that would allow us to write this proof term manually. For now, we just say that the algorithmic problem of "logically complete case analysis" is undecidable when phrased in Coq's logic. A few tactics and design patterns that we will present in this chapter suffice in almost all cases. For the current example, what we want is a tactic called %\index{tactics!inversion}%[inversion], which corresponds to the concept of inversion that is frequently used with natural deduction proof systems. *)
adamc@49 510
adamc@49 511 Undo.
adamc@49 512 inversion 1.
adamc@55 513 (* end thide *)
adamc@49 514 Qed.
adamc@49 515
adamc@49 516 (** What does [inversion] do? Think of it as a version of [destruct] that does its best to take advantage of the structure of arguments to inductive types. In this case, [inversion] completed the proof immediately, because it was able to detect that we were using [isZero] with an impossible argument.
adamc@49 517
adamc@49 518 Sometimes using [destruct] when you should have used [inversion] can lead to confusing results. To illustrate, consider an alternate proof attempt for the last theorem. *)
adamc@49 519
adamc@49 520 Theorem isZero_contra' : isZero 1 -> 2 + 2 = 5.
adamc@49 521 destruct 1.
adamc@49 522 (** [[
adamc@49 523 ============================
adamc@49 524 1 + 1 = 4
adamc@209 525
adamc@209 526 ]]
adamc@49 527
adam@280 528 What on earth happened here? Internally, [destruct] replaced [1] with a fresh variable, and, trying to be helpful, it also replaced the occurrence of [1] within the unary representation of each number in the goal. This has the net effect of decrementing each of these numbers. *)
adamc@209 529
adamc@49 530 Abort.
adamc@49 531
adam@280 532 (** To see more clearly what is happening, we can consider the type of [isZero]'s induction principle. *)
adam@280 533
adam@280 534 Check isZero_ind.
adam@280 535 (** %\vspace{-.15in}% [[
adam@280 536 isZero_ind
adam@280 537 : forall P : nat -> Prop, P 0 -> forall n : nat, isZero n -> P n
adam@280 538
adam@280 539 ]]
adam@280 540
adam@401 541 In our last proof script, [destruct] chose to instantiate [P] as [fun n => S n + S n = S (S (S (S n)))]. You can verify for yourself that this specialization of the principle applies to the goal and that the hypothesis [P 0] then matches the subgoal we saw generated. If you are doing a proof and encounter a strange transmutation like this, there is a good chance that you should go back and replace a use of [destruct] with [inversion]. *)
adam@280 542
adamc@49 543
adamc@49 544 (* begin hide *)
adamc@49 545 (* In-class exercises *)
adamc@49 546
adamc@49 547 (* EX: Define an inductive type capturing when a list has exactly two elements. Prove that your predicate does not hold of the empty list, and prove that, whenever it holds of a list, the length of that list is two. *)
adamc@49 548
adamc@52 549 (* begin thide *)
adamc@52 550 Section twoEls.
adamc@52 551 Variable A : Type.
adamc@52 552
adamc@52 553 Inductive twoEls : list A -> Prop :=
adamc@52 554 | TwoEls : forall x y, twoEls (x :: y :: nil).
adamc@52 555
adamc@52 556 Theorem twoEls_nil : twoEls nil -> False.
adamc@52 557 inversion 1.
adamc@52 558 Qed.
adamc@52 559
adamc@52 560 Theorem twoEls_two : forall ls, twoEls ls -> length ls = 2.
adamc@52 561 inversion 1.
adamc@52 562 reflexivity.
adamc@52 563 Qed.
adamc@52 564 End twoEls.
adamc@52 565 (* end thide *)
adamc@52 566
adamc@49 567 (* end hide *)
adamc@49 568
adamc@50 569
adamc@50 570 (** * Recursive Predicates *)
adamc@50 571
adamc@50 572 (** We have already seen all of the ingredients we need to build interesting recursive predicates, like this predicate capturing even-ness. *)
adamc@50 573
adamc@50 574 Inductive even : nat -> Prop :=
adamc@50 575 | EvenO : even O
adamc@50 576 | EvenSS : forall n, even n -> even (S (S n)).
adamc@50 577
adam@401 578 (** Think of [even] as another judgment defined by natural deduction rules. The rule [EvenO] has nothing above the line and [even O] below the line, and [EvenSS] is a rule with [even n] above the line and [even (S (S n))] below.
adamc@50 579
adamc@50 580 The proof techniques of the last section are easily adapted. *)
adamc@50 581
adamc@50 582 Theorem even_0 : even 0.
adamc@55 583 (* begin thide *)
adamc@50 584 constructor.
adamc@55 585 (* end thide *)
adamc@50 586 Qed.
adamc@50 587
adamc@50 588 Theorem even_4 : even 4.
adamc@55 589 (* begin thide *)
adamc@50 590 constructor; constructor; constructor.
adamc@55 591 (* end thide *)
adamc@50 592 Qed.
adamc@50 593
adam@375 594 (** It is not hard to see that sequences of constructor applications like the above can get tedious. We can avoid them using Coq's hint facility, with a new [Hint] variant that asks to consider all constructors of an inductive type during proof search. The tactic %\index{tactics!auto}%[auto] performs exhaustive proof search up to a fixed depth, considering only the proof steps we have registered as hints. *)
adamc@50 595
adamc@55 596 (* begin thide *)
adamc@50 597 Hint Constructors even.
adamc@50 598
adamc@50 599 Theorem even_4' : even 4.
adamc@50 600 auto.
adamc@50 601 Qed.
adamc@50 602
adamc@55 603 (* end thide *)
adamc@55 604
adam@322 605 (** We may also use [inversion] with [even]. *)
adam@322 606
adamc@50 607 Theorem even_1_contra : even 1 -> False.
adamc@55 608 (* begin thide *)
adamc@50 609 inversion 1.
adamc@55 610 (* end thide *)
adamc@50 611 Qed.
adamc@50 612
adamc@50 613 Theorem even_3_contra : even 3 -> False.
adamc@55 614 (* begin thide *)
adamc@50 615 inversion 1.
adamc@50 616 (** [[
adamc@50 617 H : even 3
adamc@50 618 n : nat
adamc@50 619 H1 : even 1
adamc@50 620 H0 : n = 1
adamc@50 621 ============================
adamc@50 622 False
adamc@209 623
adamc@209 624 ]]
adamc@50 625
adam@322 626 The [inversion] tactic can be a little overzealous at times, as we can see here with the introduction of the unused variable [n] and an equality hypothesis about it. For more complicated predicates, though, adding such assumptions is critical to dealing with the undecidability of general inversion. More complex inductive definitions and theorems can cause [inversion] to generate equalities where neither side is a variable. *)
adamc@50 627
adamc@50 628 inversion H1.
adamc@55 629 (* end thide *)
adamc@50 630 Qed.
adamc@50 631
adamc@50 632 (** We can also do inductive proofs about [even]. *)
adamc@50 633
adamc@50 634 Theorem even_plus : forall n m, even n -> even m -> even (n + m).
adamc@55 635 (* begin thide *)
adamc@50 636 (** It seems a reasonable first choice to proceed by induction on [n]. *)
adamc@209 637
adamc@50 638 induction n; crush.
adamc@50 639 (** [[
adamc@50 640 n : nat
adamc@50 641 IHn : forall m : nat, even n -> even m -> even (n + m)
adamc@50 642 m : nat
adamc@50 643 H : even (S n)
adamc@50 644 H0 : even m
adamc@50 645 ============================
adamc@50 646 even (S (n + m))
adamc@209 647
adamc@209 648 ]]
adamc@50 649
adamc@209 650 We will need to use the hypotheses [H] and [H0] somehow. The most natural choice is to invert [H]. *)
adamc@50 651
adamc@50 652 inversion H.
adamc@50 653 (** [[
adamc@50 654 n : nat
adamc@50 655 IHn : forall m : nat, even n -> even m -> even (n + m)
adamc@50 656 m : nat
adamc@50 657 H : even (S n)
adamc@50 658 H0 : even m
adamc@50 659 n0 : nat
adamc@50 660 H2 : even n0
adamc@50 661 H1 : S n0 = n
adamc@50 662 ============================
adamc@50 663 even (S (S n0 + m))
adamc@209 664
adamc@209 665 ]]
adamc@50 666
adamc@209 667 Simplifying the conclusion brings us to a point where we can apply a constructor. *)
adamc@209 668
adamc@50 669 simpl.
adamc@50 670 (** [[
adamc@50 671 ============================
adamc@50 672 even (S (S (n0 + m)))
adam@302 673 ]]
adam@302 674 *)
adamc@50 675
adamc@50 676 constructor.
adam@322 677
adam@401 678 (** [[
adamc@50 679 ============================
adamc@50 680 even (n0 + m)
adamc@209 681
adamc@209 682 ]]
adamc@50 683
adamc@209 684 At this point, we would like to apply the inductive hypothesis, which is:
adamc@209 685
adamc@209 686 [[
adamc@50 687 IHn : forall m : nat, even n -> even m -> even (n + m)
adamc@209 688 ]]
adamc@50 689
adam@421 690 Unfortunately, the goal mentions [n0] where it would need to mention [n] to match [IHn]. We could keep looking for a way to finish this proof from here, but it turns out that we can make our lives much easier by changing our basic strategy. Instead of inducting on the structure of [n], we should induct _on the structure of one of the [even] proofs_. This technique is commonly called%\index{rule induction}% _rule induction_ in programming language semantics. In the setting of Coq, we have already seen how predicates are defined using the same inductive type mechanism as datatypes, so the fundamental unity of rule induction with "normal" induction is apparent.
adamc@50 691
adam@322 692 Recall that tactics like [induction] and [destruct] may be passed numbers to refer to unnamed lefthand sides of implications in the conclusion, where the argument [n] refers to the [n]th such hypothesis. *)
adam@322 693
adamc@50 694 Restart.
adamc@50 695
adamc@50 696 induction 1.
adamc@50 697 (** [[
adamc@50 698 m : nat
adamc@50 699 ============================
adamc@50 700 even m -> even (0 + m)
adam@322 701 ]]
adamc@50 702
adam@322 703 %\noindent \coqdockw{subgoal} 2 \coqdockw{is}:%#<tt>subgoal 2 is</tt>#
adam@322 704 [[
adamc@50 705 even m -> even (S (S n) + m)
adamc@209 706
adamc@209 707 ]]
adamc@50 708
adamc@209 709 The first case is easily discharged by [crush], based on the hint we added earlier to try the constructors of [even]. *)
adamc@50 710
adamc@50 711 crush.
adamc@50 712
adamc@50 713 (** Now we focus on the second case: *)
adamc@209 714
adamc@50 715 intro.
adamc@50 716 (** [[
adamc@50 717 m : nat
adamc@50 718 n : nat
adamc@50 719 H : even n
adamc@50 720 IHeven : even m -> even (n + m)
adamc@50 721 H0 : even m
adamc@50 722 ============================
adamc@50 723 even (S (S n) + m)
adamc@209 724
adamc@209 725 ]]
adamc@50 726
adamc@209 727 We simplify and apply a constructor, as in our last proof attempt. *)
adamc@50 728
adamc@50 729 simpl; constructor.
adam@322 730
adam@401 731 (** [[
adamc@50 732 ============================
adamc@50 733 even (n + m)
adamc@209 734
adamc@209 735 ]]
adamc@50 736
adamc@209 737 Now we have an exact match with our inductive hypothesis, and the remainder of the proof is trivial. *)
adamc@50 738
adamc@50 739 apply IHeven; assumption.
adamc@50 740
adamc@50 741 (** In fact, [crush] can handle all of the details of the proof once we declare the induction strategy. *)
adamc@50 742
adamc@50 743 Restart.
adam@322 744
adamc@50 745 induction 1; crush.
adamc@55 746 (* end thide *)
adamc@50 747 Qed.
adamc@50 748
adamc@50 749 (** Induction on recursive predicates has similar pitfalls to those we encountered with inversion in the last section. *)
adamc@50 750
adamc@50 751 Theorem even_contra : forall n, even (S (n + n)) -> False.
adamc@55 752 (* begin thide *)
adamc@50 753 induction 1.
adamc@50 754 (** [[
adamc@50 755 n : nat
adamc@50 756 ============================
adamc@50 757 False
adam@322 758 ]]
adamc@50 759
adam@322 760 %\noindent \coqdockw{subgoal} 2 \coqdockw{is}:%#<tt>subgoal 2 is</tt>#
adam@322 761 [[
adamc@50 762 False
adamc@209 763
adamc@209 764 ]]
adamc@50 765
adam@280 766 We are already sunk trying to prove the first subgoal, since the argument to [even] was replaced by a fresh variable internally. This time, we find it easier to prove this theorem by way of a lemma. Instead of trusting [induction] to replace expressions with fresh variables, we do it ourselves, explicitly adding the appropriate equalities as new assumptions. *)
adamc@209 767
adamc@50 768 Abort.
adamc@50 769
adamc@50 770 Lemma even_contra' : forall n', even n' -> forall n, n' = S (n + n) -> False.
adamc@50 771 induction 1; crush.
adamc@50 772
adamc@54 773 (** At this point, it is useful to consider all cases of [n] and [n0] being zero or nonzero. Only one of these cases has any trickiness to it. *)
adamc@209 774
adamc@50 775 destruct n; destruct n0; crush.
adamc@50 776
adamc@50 777 (** [[
adamc@50 778 n : nat
adamc@50 779 H : even (S n)
adamc@50 780 IHeven : forall n0 : nat, S n = S (n0 + n0) -> False
adamc@50 781 n0 : nat
adamc@50 782 H0 : S n = n0 + S n0
adamc@50 783 ============================
adamc@50 784 False
adamc@209 785
adamc@209 786 ]]
adamc@50 787
adam@280 788 At this point it is useful to use a theorem from the standard library, which we also proved with a different name in the last chapter. We can search for a theorem that allows us to rewrite terms of the form [x + S y]. *)
adamc@209 789
adam@280 790 SearchRewrite (_ + S _).
adam@322 791
adam@401 792 (** [[
adam@280 793 plus_n_Sm : forall n m : nat, S (n + m) = n + S m
adam@302 794 ]]
adam@302 795 *)
adamc@50 796
adamc@50 797 rewrite <- plus_n_Sm in H0.
adamc@50 798
adam@322 799 (** The induction hypothesis lets us complete the proof, if we use a variant of [apply] that has a %\index{tactics!with}%[with] clause to give instantiations of quantified variables. *)
adamc@209 800
adamc@50 801 apply IHeven with n0; assumption.
adamc@50 802
adam@322 803 (** As usual, we can rewrite the proof to avoid referencing any locally generated names, which makes our proof script more readable and more robust to changes in the theorem statement. We use the notation [<-] to request a hint that does right-to-left rewriting, just like we can with the [rewrite] tactic. *)
adamc@209 804
adamc@209 805 Restart.
adam@322 806
adam@375 807 Hint Rewrite <- plus_n_Sm.
adamc@50 808
adamc@50 809 induction 1; crush;
adamc@50 810 match goal with
adamc@50 811 | [ H : S ?N = ?N0 + ?N0 |- _ ] => destruct N; destruct N0
adamc@50 812 end; crush; eauto.
adamc@50 813 Qed.
adamc@50 814
adam@322 815 (** We write the proof in a way that avoids the use of local variable or hypothesis names, using the %\index{tactics!match}%[match] tactic form to do pattern-matching on the goal. We use unification variables prefixed by question marks in the pattern, and we take advantage of the possibility to mention a unification variable twice in one pattern, to enforce equality between occurrences. The hint to rewrite with [plus_n_Sm] in a particular direction saves us from having to figure out the right place to apply that theorem, and we also take critical advantage of a new tactic, %\index{tactics!eauto}%[eauto].
adamc@50 816
adam@421 817 The [crush] tactic uses the tactic [intuition], which, when it runs out of tricks to try using only propositional logic, by default tries the tactic [auto], which we saw in an earlier example. The [auto] tactic attempts %\index{Prolog}%Prolog-style logic programming, searching through all proof trees up to a certain depth that are built only out of hints that have been registered with [Hint] commands. Compared to Prolog, [auto] places an important restriction: it never introduces new unification variables during search. That is, every time a rule is applied during proof search, all of its arguments must be deducible by studying the form of the goal. This restriction is relaxed for [eauto], at the cost of possibly exponentially greater running time. In this particular case, we know that [eauto] has only a small space of proofs to search, so it makes sense to run it. It is common in effectively automated Coq proofs to see a bag of standard tactics applied to pick off the "easy" subgoals, finishing with [eauto] to handle the tricky parts that can benefit from ad-hoc exhaustive search.
adamc@50 818
adamc@50 819 The original theorem now follows trivially from our lemma. *)
adamc@50 820
adamc@50 821 Theorem even_contra : forall n, even (S (n + n)) -> False.
adamc@52 822 intros; eapply even_contra'; eauto.
adamc@50 823 Qed.
adamc@52 824
adam@398 825 (** We use a variant %\index{tactics!apply}%[eapply] of [apply] which has the same relationship to [apply] as [eauto] has to [auto]. An invocation of [apply] only succeeds if all arguments to the rule being used can be determined from the form of the goal, whereas [eapply] will introduce unification variables for undetermined arguments. In this case, [eauto] is able to determine the right values for those unification variables, using (unsurprisingly) a variant of the classic algorithm for _unification_ %\cite{unification}%.
adamc@52 826
adamc@52 827 By considering an alternate attempt at proving the lemma, we can see another common pitfall of inductive proofs in Coq. Imagine that we had tried to prove [even_contra'] with all of the [forall] quantifiers moved to the front of the lemma statement. *)
adamc@52 828
adamc@52 829 Lemma even_contra'' : forall n' n, even n' -> n' = S (n + n) -> False.
adamc@52 830 induction 1; crush;
adamc@52 831 match goal with
adamc@52 832 | [ H : S ?N = ?N0 + ?N0 |- _ ] => destruct N; destruct N0
adamc@52 833 end; crush; eauto.
adamc@52 834
adamc@209 835 (** One subgoal remains:
adamc@52 836
adamc@209 837 [[
adamc@52 838 n : nat
adamc@52 839 H : even (S (n + n))
adamc@52 840 IHeven : S (n + n) = S (S (S (n + n))) -> False
adamc@52 841 ============================
adamc@52 842 False
adamc@209 843
adamc@209 844 ]]
adamc@52 845
adam@398 846 We are out of luck here. The inductive hypothesis is trivially true, since its assumption is false. In the version of this proof that succeeded, [IHeven] had an explicit quantification over [n]. This is because the quantification of [n] _appeared after the thing we are inducting on_ in the theorem statement. In general, quantified variables and hypotheses that appear before the induction object in the theorem statement stay fixed throughout the inductive proof. Variables and hypotheses that are quantified after the induction object may be varied explicitly in uses of inductive hypotheses. *)
adamc@52 847
adam@322 848 Abort.
adam@322 849
adam@322 850 (** Why should Coq implement [induction] this way? One answer is that it avoids burdening this basic tactic with additional heuristic smarts, but that is not the whole picture. Imagine that [induction] analyzed dependencies among variables and reordered quantifiers to preserve as much freedom as possible in later uses of inductive hypotheses. This could make the inductive hypotheses more complex, which could in turn cause particular automation machinery to fail when it would have succeeded before. In general, we want to avoid quantifiers in our proofs whenever we can, and that goal is furthered by the refactoring that the [induction] tactic forces us to do. *)
adamc@55 851 (* end thide *)
adamc@209 852
adam@322 853
adamc@51 854
adamc@52 855
adamc@52 856 (* begin hide *)
adamc@52 857 (* In-class exercises *)
adamc@52 858
adam@421 859 (* EX: Define a type [prop] of simple boolean formulas made up only of truth, falsehood, binary conjunction, and binary disjunction. Define an inductive predicate [holds] that captures when [prop]s are valid, and define a predicate [falseFree] that captures when a [prop] does not contain the "false" formula. Prove that every false-free [prop] is valid. *)
adamc@52 860
adamc@52 861 (* begin thide *)
adamc@52 862 Inductive prop : Set :=
adamc@52 863 | Tru : prop
adamc@52 864 | Fals : prop
adamc@52 865 | And : prop -> prop -> prop
adamc@52 866 | Or : prop -> prop -> prop.
adamc@52 867
adamc@52 868 Inductive holds : prop -> Prop :=
adamc@52 869 | HTru : holds Tru
adamc@52 870 | HAnd : forall p1 p2, holds p1 -> holds p2 -> holds (And p1 p2)
adamc@52 871 | HOr1 : forall p1 p2, holds p1 -> holds (Or p1 p2)
adamc@52 872 | HOr2 : forall p1 p2, holds p2 -> holds (Or p1 p2).
adamc@52 873
adamc@52 874 Inductive falseFree : prop -> Prop :=
adamc@52 875 | FFTru : falseFree Tru
adamc@52 876 | FFAnd : forall p1 p2, falseFree p1 -> falseFree p2 -> falseFree (And p1 p2)
adamc@52 877 | FFNot : forall p1 p2, falseFree p1 -> falseFree p2 -> falseFree (Or p1 p2).
adamc@52 878
adamc@52 879 Hint Constructors holds.
adamc@52 880
adamc@52 881 Theorem falseFree_holds : forall p, falseFree p -> holds p.
adamc@52 882 induction 1; crush.
adamc@52 883 Qed.
adamc@52 884 (* end thide *)
adamc@52 885
adamc@52 886
adamc@52 887 (* EX: Define an inductive type [prop'] that is the same as [prop] but omits the possibility for falsehood. Define a proposition [holds'] for [prop'] that is analogous to [holds]. Define a function [propify] for translating [prop']s to [prop]s. Prove that, for any [prop'] [p], if [propify p] is valid, then so is [p]. *)
adamc@52 888
adamc@52 889 (* begin thide *)
adamc@52 890 Inductive prop' : Set :=
adamc@52 891 | Tru' : prop'
adamc@52 892 | And' : prop' -> prop' -> prop'
adamc@52 893 | Or' : prop' -> prop' -> prop'.
adamc@52 894
adamc@52 895 Inductive holds' : prop' -> Prop :=
adamc@52 896 | HTru' : holds' Tru'
adamc@52 897 | HAnd' : forall p1 p2, holds' p1 -> holds' p2 -> holds' (And' p1 p2)
adamc@52 898 | HOr1' : forall p1 p2, holds' p1 -> holds' (Or' p1 p2)
adamc@52 899 | HOr2' : forall p1 p2, holds' p2 -> holds' (Or' p1 p2).
adamc@52 900
adamc@52 901 Fixpoint propify (p : prop') : prop :=
adamc@52 902 match p with
adamc@52 903 | Tru' => Tru
adamc@52 904 | And' p1 p2 => And (propify p1) (propify p2)
adamc@52 905 | Or' p1 p2 => Or (propify p1) (propify p2)
adamc@52 906 end.
adamc@52 907
adamc@52 908 Hint Constructors holds'.
adamc@52 909
adamc@52 910 Lemma propify_holds' : forall p', holds p' -> forall p, p' = propify p -> holds' p.
adamc@52 911 induction 1; crush; destruct p; crush.
adamc@52 912 Qed.
adamc@52 913
adamc@52 914 Theorem propify_holds : forall p, holds (propify p) -> holds' p.
adamc@52 915 intros; eapply propify_holds'; eauto.
adamc@52 916 Qed.
adamc@52 917 (* end thide *)
adamc@52 918
adamc@52 919 (* end hide *)