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Typesetting pass over GeneralRec
author Adam Chlipala <adam@chlipala.net>
date Fri, 08 Jun 2012 13:59:41 -0400
parents d27c0be2c3d4
children 539ed97750bb
rev   line source
adam@398 1 (* Copyright (c) 2008-2012, 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
adam@314 13 Require Import CpdtTactics.
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@315 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.
adam@315 24
adam@315 25 The last chapter took a deep dive into some of the more advanced Coq features, to highlight the unusual approach that I advocate in this book. However, from this point on, we will rewind and go back to basics, presenting the relevant features of Coq in a more bottom-up manner. A useful first step is a discussion of the differences and relationships between proofs and programs in Coq. *)
adam@315 26
adam@315 27
adam@315 28 (** * Proof Terms *)
adam@315 29
adam@315 30 (** Mainstream presentations of mathematics treat proofs as objects that exist outside of the universe of mathematical objects. However, for a variety of reasoning, it is convenient to encode proofs, traditional mathematical objects, and programs within a single formal language. Validity checks on mathematical objects are useful in any setting, to catch typoes and other uninteresting errors. The benefits of static typing for programs are widely recognized, and Coq brings those benefits to both mathematical objects and programs via a uniform mechanism. In fact, from this point on, we will not bother to distinguish between programs and mathematical objects. Many mathematical formalisms are most easily encoded in terms of programs.
adam@315 31
adam@400 32 Proofs are fundamentally different from programs, because any two proofs of a theorem are considered equivalent, from a formal standpoint if not from an engineering standpoint. However, we can use the same type-checking technology to check proofs as we use to validate our programs. This is the%\index{Curry-Howard correspondence}% _Curry-Howard correspondence_ %\cite{Curry,Howard}%, an approach for relating proofs and programs. We represent mathematical theorems as types, such that a theorem's proofs are exactly those programs that type-check at the corresponding type.
adam@315 33
adam@315 34 The last chapter's example already snuck in an instance of Curry-Howard. We used the token [->] to stand for both function types and logical implications. One reasonable conclusion upon seeing this might be that some fancy overloading of notations is at work. In fact, functions and implications are precisely identical according to Curry-Howard! That is, they are just two ways of describing the same computational phenomenon.
adam@315 35
adam@315 36 A short demonstration should explain how this can be. The identity function over the natural numbers is certainly not a controversial program. *)
adam@315 37
adam@315 38 Check (fun x : nat => x).
adam@315 39 (** [: nat -> nat] *)
adam@315 40
adam@315 41 (** Consider this alternate program, which is almost identical to the last one. *)
adam@315 42
adam@315 43 Check (fun x : True => x).
adam@315 44 (** [: True -> True] *)
adam@315 45
adam@315 46 (** The identity program is interpreted as a proof that %\index{Gallina terms!True}%[True], the always-true proposition, implies itself! What we see is that Curry-Howard interprets implications as functions, where an input is a proposition being assumed and an output is a proposition being deduced. This intuition is not too far from a common one for informal theorem proving, where we might already think of an implication proof as a process for transforming a hypothesis into a conclusion.
adam@315 47
adam@315 48 There are also more primitive proof forms available. For instance, the term %\index{Gallina terms!I}%[I] is the single proof of [True], applicable in any context. *)
adam@315 49
adam@315 50 Check I.
adam@315 51 (** [: True] *)
adam@315 52
adam@315 53 (** With [I], we can prove another simple propositional theorem. *)
adam@315 54
adam@315 55 Check (fun _ : False => I).
adam@315 56 (** [: False -> True] *)
adam@315 57
adam@315 58 (** No proofs of %\index{Gallina terms!False}%[False] exist in the top-level context, but the implication-as-function analogy gives us an easy way to, for example, show that [False] implies itself. *)
adam@315 59
adam@315 60 Check (fun x : False => x).
adam@315 61 (** [: False -> False] *)
adam@315 62
adam@315 63 (** In fact, [False] implies anything, and we can take advantage of this fact with an odd looking [match] expression that has no branches. Since there are no rules for deducing [False], there are no cases to consider! *)
adam@315 64
adam@315 65 Check (fun x : False => match x with end : True).
adam@315 66 (** [: False -> True] *)
adam@315 67
adam@398 68 (** Every one of these example programs whose type looks like a logical formula is a %\index{proof term}%_proof term_. We use that name for any Gallina term of a logical type, and we will elaborate shortly on what makes a type logical.
adam@315 69
adam@317 70 In the rest of this chapter, we will introduce different ways of defining types. Every example type can be interpreted alternatively as a type of programs or %\index{proposition}%propositions (i.e., formulas or theorem statements).
adam@317 71
adam@398 72 One of the first types we introduce will be [bool], with constructors [true] and [false]. Newcomers to Coq often wonder about the distinction between [True] and [true] and the distinction between [False] and [false]. One glib answer is that [True] and [False] are types, but [true] and [false] are not. A more useful answer is that Coq's metatheory guarantees that any term of type [bool] _evaluates_ to either [true] or [false]. This means that we have an _algorithm_ for answering any question phrased as an expression of type [bool]. Conversely, most propositions do not evaluate to [True] or [False]; the language of inductively defined propositions is much richer than that. We ought to be glad that we have no algorithm for deciding mathematical truth, since otherwise it would be clear that we could not formalize undecidable properties, like most any properties of general-purpose programs. *)
adamc@26 73
adamc@26 74
adamc@26 75 (** * Enumerations *)
adamc@26 76
adam@400 77 (** Coq inductive types generalize the %\index{algebraic datatypes}%algebraic datatypes found in %\index{Haskell}%Haskell and %\index{ML}%ML. Confusingly enough, inductive types also generalize %\index{generalized algebraic datatypes}%generalized algebraic datatypes (GADT's), 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 78
adam@315 79 The singleton type [unit] is an inductive type:%\index{Gallina terms!unit}\index{Gallina terms!tt}% *)
adamc@26 80
adamc@26 81 Inductive unit : Set :=
adamc@26 82 | tt.
adamc@26 83
adamc@26 84 (** 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 85
adamc@26 86 Check unit.
adamc@208 87 (** [unit : Set] *)
adamc@26 88
adamc@26 89 Check tt.
adamc@208 90 (** [tt : unit] *)
adamc@26 91
adamc@26 92 (** We can prove that [unit] is a genuine singleton type. *)
adamc@26 93
adamc@26 94 Theorem unit_singleton : forall x : unit, x = tt.
adamc@208 95
adam@315 96 (** 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].%\index{tactics!induction}% *)
adamc@208 97
adamc@41 98 (* begin thide *)
adamc@26 99 induction x.
adamc@26 100
adamc@208 101 (** The goal changes to:
adamc@208 102 [[
adamc@26 103 tt = tt
adam@302 104 ]]
adam@302 105 *)
adamc@208 106
adamc@26 107 (** ...which we can discharge trivially. *)
adamc@208 108
adamc@26 109 reflexivity.
adamc@26 110 Qed.
adamc@41 111 (* end thide *)
adamc@26 112
adam@315 113 (** 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:%\index{tactics!destruct}% [[
adamc@26 114
adamc@26 115 destruct x.
adamc@205 116
adamc@205 117 ]]
adamc@205 118
adam@292 119 %\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@26 120
adam@398 121 What exactly _is_ the %\index{induction principles}%induction principle for [unit]? We can ask Coq: *)
adamc@26 122
adamc@26 123 Check unit_ind.
adamc@208 124 (** [unit_ind : forall P : unit -> Prop, P tt -> forall u : unit, P u] *)
adamc@26 125
adam@315 126 (** Every [Inductive] command defining a type [T] also defines an induction principle named [T_ind]. Recall from the last section that 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 %\index{Gallina terms!Prop}%[Prop], which appears in our induction principle; and the type %\index{Gallina terms!Set}%[Set], which we have seen a few times already.
adamc@26 127
adam@315 128 The convention goes like this: [Set] is the type of normal types used in programming, 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@26 129
adam@400 130 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)].
adam@315 131
adam@315 132 The definition of [unit] places the type in [Set]. By replacing [Set] with [Prop], [unit] with [True], and [tt] with [I], we arrive at precisely the definition of [True] that the Coq standard library employs! The program type [unit] is the Curry-Howard equivalent of the proposition [True]. We might make the tongue-in-cheek claim that, while philosophers have expended much ink on the nature of truth, we have now determined that truth is the [unit] type of functional programming.
adamc@26 133
adamc@26 134 %\medskip%
adamc@26 135
adam@315 136 We can define an inductive type even simpler than [unit]:%\index{Gallina terms!Empty\_set}% *)
adamc@26 137
adamc@26 138 Inductive Empty_set : Set := .
adamc@26 139
adamc@26 140 (** [Empty_set] has no elements. We can prove fun theorems about it: *)
adamc@26 141
adamc@26 142 Theorem the_sky_is_falling : forall x : Empty_set, 2 + 2 = 5.
adamc@41 143 (* begin thide *)
adamc@26 144 destruct 1.
adamc@26 145 Qed.
adamc@41 146 (* end thide *)
adamc@26 147
adam@317 148 (** 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. At least within Coq's logical foundation of %\index{constructive logic}%constructive logic, which we elaborate on more in the next chapter, 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@26 149
adamc@26 150 We can see the induction principle that made this proof so easy: *)
adamc@26 151
adamc@26 152 Check Empty_set_ind.
adam@400 153 (** [Empty_set_ind : forall (P : Empty_set -> Prop) (e : Empty_set), P e] *)
adamc@26 154
adam@400 155 (** 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@26 156
adamc@26 157 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@26 158
adamc@26 159 Definition e2u (e : Empty_set) : unit := match e with end.
adamc@26 160
adam@315 161 (** 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. This idiom may look familiar; we employed it with proofs of [False] in the last section. In fact, [Empty_set] is the Curry-Howard equivalent of [False]. As for why [Empty_set] starts with a capital letter and not a lowercase letter like [unit] does, we must refer the reader to the authors of the Coq standard library, to which we try to be faithful.
adamc@26 162
adamc@26 163 %\medskip%
adamc@26 164
adam@315 165 Moving up the ladder of complexity, we can define the booleans:%\index{Gallina terms!bool}\index{Gallina terms!true}\index{Gallina terms!false}% *)
adamc@26 166
adamc@26 167 Inductive bool : Set :=
adamc@26 168 | true
adamc@26 169 | false.
adamc@26 170
adam@316 171 (** We can use less vacuous pattern matching to define boolean negation.%\index{Gallina terms!negb}% *)
adamc@26 172
adam@279 173 Definition negb (b : bool) : bool :=
adamc@26 174 match b with
adamc@26 175 | true => false
adamc@26 176 | false => true
adamc@26 177 end.
adamc@26 178
adam@317 179 (** An alternative definition desugars to the above, thanks to an %\index{Gallina terms!if}%[if] notation overloaded to work with any inductive type that has exactly two constructors: *)
adamc@27 180
adam@279 181 Definition negb' (b : bool) : bool :=
adamc@27 182 if b then false else true.
adamc@27 183
adam@279 184 (** We might want to prove that [negb] is its own inverse operation. *)
adamc@26 185
adam@279 186 Theorem negb_inverse : forall b : bool, negb (negb b) = b.
adamc@41 187 (* begin thide *)
adamc@26 188 destruct b.
adamc@26 189
adamc@208 190 (** After we case-analyze on [b], we are left with one subgoal for each constructor of [bool].
adamc@26 191
adam@315 192 %\vspace{.1in} \noindent 2 \coqdockw{subgoals}\vspace{-.1in}%#<tt>2 subgoals</tt>#
adam@315 193
adamc@26 194 [[
adamc@26 195 ============================
adam@279 196 negb (negb true) = true
adamc@26 197 ]]
adam@315 198 %\noindent \coqdockw{subgoal} 2 \coqdockw{is}:%#<tt>subgoal 2 is</tt>#
adamc@26 199 [[
adam@279 200 negb (negb false) = false
adamc@208 201
adamc@26 202 ]]
adamc@26 203
adamc@26 204 The first subgoal follows by Coq's rules of computation, so we can dispatch it easily: *)
adamc@26 205
adamc@26 206 reflexivity.
adamc@26 207
adam@315 208 (** Likewise for the second subgoal, so we can restart the proof and give a very compact justification.%\index{Vernacular commands!Restart}% *)
adamc@26 209
adamc@26 210 Restart.
adam@315 211
adamc@26 212 destruct b; reflexivity.
adamc@26 213 Qed.
adamc@41 214 (* end thide *)
adamc@27 215
adam@315 216 (** Another theorem about booleans illustrates another useful tactic.%\index{tactics!discriminate}% *)
adamc@27 217
adam@279 218 Theorem negb_ineq : forall b : bool, negb b <> b.
adamc@41 219 (* begin thide *)
adamc@27 220 destruct b; discriminate.
adamc@27 221 Qed.
adamc@41 222 (* end thide *)
adamc@27 223
adamc@27 224 (** [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@27 225
adamc@27 226 At this point, it is probably not hard to guess what the underlying induction principle for [bool] is. *)
adamc@27 227
adamc@27 228 Check bool_ind.
adamc@208 229 (** [bool_ind : forall P : bool -> Prop, P true -> P false -> forall b : bool, P b] *)
adamc@28 230
adam@315 231 (** That is, to prove that a property describes all [bool]s, prove that it describes both [true] and [false].
adam@315 232
adam@392 233 There is no interesting Curry-Howard analogue of [bool]. Of course, we can define such a type by replacing [Set] by [Prop] above, but the proposition we arrive at is not very useful. It is logically equivalent to [True], but it provides two indistinguishable primitive proofs, [true] and [false]. In the rest of the chapter, we will skip commenting on Curry-Howard versions of inductive definitions where such versions are not interesting. *)
adam@315 234
adamc@28 235
adamc@28 236 (** * Simple Recursive Types *)
adamc@28 237
adam@315 238 (** The natural numbers are the simplest common example of an inductive type that actually deserves the name.%\index{Gallina terms!nat}\index{Gallina terms!O}\index{Gallina terms!S}% *)
adamc@28 239
adamc@28 240 Inductive nat : Set :=
adamc@28 241 | O : nat
adamc@28 242 | S : nat -> nat.
adamc@28 243
adam@400 244 (** [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@28 245
adam@316 246 Pattern matching works as we demonstrated in the last chapter:%\index{Gallina terms!pred}% *)
adamc@28 247
adamc@28 248 Definition isZero (n : nat) : bool :=
adamc@28 249 match n with
adamc@28 250 | O => true
adamc@28 251 | S _ => false
adamc@28 252 end.
adamc@28 253
adamc@28 254 Definition pred (n : nat) : nat :=
adamc@28 255 match n with
adamc@28 256 | O => O
adamc@28 257 | S n' => n'
adamc@28 258 end.
adamc@28 259
adamc@28 260 (** We can prove theorems by case analysis: *)
adamc@28 261
adamc@28 262 Theorem S_isZero : forall n : nat, isZero (pred (S (S n))) = false.
adamc@41 263 (* begin thide *)
adamc@28 264 destruct n; reflexivity.
adamc@28 265 Qed.
adamc@41 266 (* end thide *)
adamc@28 267
adam@316 268 (** We can also now get into genuine inductive theorems. First, we will need a recursive function, to make things interesting.%\index{Gallina terms!plus}% *)
adamc@28 269
adamc@208 270 Fixpoint plus (n m : nat) : nat :=
adamc@28 271 match n with
adamc@28 272 | O => m
adamc@28 273 | S n' => S (plus n' m)
adamc@28 274 end.
adamc@28 275
adamc@208 276 (** Recall that [Fixpoint] is Coq's mechanism for recursive function definitions. Some theorems about [plus] can be proved without induction. *)
adamc@28 277
adamc@28 278 Theorem O_plus_n : forall n : nat, plus O n = n.
adamc@41 279 (* begin thide *)
adamc@28 280 intro; reflexivity.
adamc@28 281 Qed.
adamc@41 282 (* end thide *)
adamc@28 283
adamc@208 284 (** 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@28 285
adamc@28 286 Theorem n_plus_O : forall n : nat, plus n O = n.
adamc@41 287 (* begin thide *)
adamc@28 288 induction n.
adamc@28 289
adam@398 290 (** Our first subgoal is [plus O O = O], which _is_ trivial by computation. *)
adamc@28 291
adamc@28 292 reflexivity.
adamc@28 293
adamc@28 294 (** Our second subgoal is more work and also demonstrates our first inductive hypothesis.
adamc@28 295
adamc@28 296 [[
adamc@28 297 n : nat
adamc@28 298 IHn : plus n O = n
adamc@28 299 ============================
adamc@28 300 plus (S n) O = S n
adamc@208 301
adamc@28 302 ]]
adamc@28 303
adam@315 304 We can start out by using computation to simplify the goal as far as we can.%\index{tactics!simpl}% *)
adamc@28 305
adamc@28 306 simpl.
adamc@28 307
adam@400 308 (** Now the conclusion is [S (plus n O) = S n]. Using our inductive hypothesis: *)
adamc@28 309
adamc@28 310 rewrite IHn.
adamc@28 311
adamc@28 312 (** ...we get a trivial conclusion [S n = S n]. *)
adamc@28 313
adamc@28 314 reflexivity.
adamc@28 315
adam@315 316 (** Not much really went on in this proof, so the [crush] tactic from the [CpdtTactics] module can prove this theorem automatically. *)
adamc@28 317
adamc@28 318 Restart.
adam@315 319
adamc@28 320 induction n; crush.
adamc@28 321 Qed.
adamc@41 322 (* end thide *)
adamc@28 323
adamc@28 324 (** We can check out the induction principle at work here: *)
adamc@28 325
adamc@28 326 Check nat_ind.
adamc@208 327 (** %\vspace{-.15in}% [[
adamc@208 328 nat_ind : forall P : nat -> Prop,
adamc@208 329 P O -> (forall n : nat, P n -> P (S n)) -> forall n : nat, P n
adamc@208 330
adamc@208 331 ]]
adamc@28 332
adam@400 333 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@28 334
adam@315 335 Since [nat] has a constructor that takes an argument, we may sometimes need to know that that constructor is injective.%\index{tactics!injection}\index{tactics!trivial}% *)
adamc@28 336
adamc@28 337 Theorem S_inj : forall n m : nat, S n = S m -> n = m.
adamc@41 338 (* begin thide *)
adamc@28 339 injection 1; trivial.
adamc@28 340 Qed.
adamc@41 341 (* end thide *)
adamc@28 342
adam@400 343 (** The [injection] tactic 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@28 344
adam@400 345 There is also a very useful tactic called %\index{tactics!congruence}%[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%\index{theory of equality and uninterpreted functions}% _complete decision procedure for the theory of equality and uninterpreted functions_, plus some smarts about inductive types.
adamc@29 346
adamc@29 347 %\medskip%
adamc@29 348
adamc@29 349 We can define a type of lists of natural numbers. *)
adamc@29 350
adamc@29 351 Inductive nat_list : Set :=
adamc@29 352 | NNil : nat_list
adamc@29 353 | NCons : nat -> nat_list -> nat_list.
adamc@29 354
adamc@29 355 (** Recursive definitions are straightforward extensions of what we have seen before. *)
adamc@29 356
adamc@29 357 Fixpoint nlength (ls : nat_list) : nat :=
adamc@29 358 match ls with
adamc@29 359 | NNil => O
adamc@29 360 | NCons _ ls' => S (nlength ls')
adamc@29 361 end.
adamc@29 362
adamc@208 363 Fixpoint napp (ls1 ls2 : nat_list) : nat_list :=
adamc@29 364 match ls1 with
adamc@29 365 | NNil => ls2
adamc@29 366 | NCons n ls1' => NCons n (napp ls1' ls2)
adamc@29 367 end.
adamc@29 368
adamc@29 369 (** Inductive theorem proving can again be automated quite effectively. *)
adamc@29 370
adamc@29 371 Theorem nlength_napp : forall ls1 ls2 : nat_list, nlength (napp ls1 ls2)
adamc@29 372 = plus (nlength ls1) (nlength ls2).
adamc@41 373 (* begin thide *)
adamc@29 374 induction ls1; crush.
adamc@29 375 Qed.
adamc@41 376 (* end thide *)
adamc@29 377
adamc@29 378 Check nat_list_ind.
adamc@208 379 (** %\vspace{-.15in}% [[
adamc@208 380 nat_list_ind
adamc@29 381 : forall P : nat_list -> Prop,
adamc@29 382 P NNil ->
adamc@29 383 (forall (n : nat) (n0 : nat_list), P n0 -> P (NCons n n0)) ->
adamc@29 384 forall n : nat_list, P n
adamc@29 385 ]]
adamc@29 386
adamc@29 387 %\medskip%
adamc@29 388
adam@292 389 In general, we can implement any %``%#"#tree#"#%''% types as inductive types. For example, here are binary trees of naturals. *)
adamc@29 390
adamc@29 391 Inductive nat_btree : Set :=
adamc@29 392 | NLeaf : nat_btree
adamc@29 393 | NNode : nat_btree -> nat -> nat_btree -> nat_btree.
adamc@29 394
adamc@29 395 Fixpoint nsize (tr : nat_btree) : nat :=
adamc@29 396 match tr with
adamc@35 397 | NLeaf => S O
adamc@29 398 | NNode tr1 _ tr2 => plus (nsize tr1) (nsize tr2)
adamc@29 399 end.
adamc@29 400
adamc@208 401 Fixpoint nsplice (tr1 tr2 : nat_btree) : nat_btree :=
adamc@29 402 match tr1 with
adamc@35 403 | NLeaf => NNode tr2 O NLeaf
adamc@29 404 | NNode tr1' n tr2' => NNode (nsplice tr1' tr2) n tr2'
adamc@29 405 end.
adamc@29 406
adamc@29 407 Theorem plus_assoc : forall n1 n2 n3 : nat, plus (plus n1 n2) n3 = plus n1 (plus n2 n3).
adamc@41 408 (* begin thide *)
adamc@29 409 induction n1; crush.
adamc@29 410 Qed.
adamc@41 411 (* end thide *)
adamc@29 412
adamc@29 413 Theorem nsize_nsplice : forall tr1 tr2 : nat_btree, nsize (nsplice tr1 tr2)
adamc@29 414 = plus (nsize tr2) (nsize tr1).
adamc@41 415 (* begin thide *)
adam@375 416 Hint Rewrite n_plus_O plus_assoc.
adamc@29 417
adamc@29 418 induction tr1; crush.
adamc@29 419 Qed.
adamc@41 420 (* end thide *)
adamc@29 421
adam@315 422 (** It is convenient that these proofs go through so easily, but it is useful to check that the tree induction principle works as usual. *)
adam@315 423
adamc@29 424 Check nat_btree_ind.
adamc@208 425 (** %\vspace{-.15in}% [[
adamc@208 426 nat_btree_ind
adamc@29 427 : forall P : nat_btree -> Prop,
adamc@29 428 P NLeaf ->
adamc@29 429 (forall n : nat_btree,
adamc@29 430 P n -> forall (n0 : nat) (n1 : nat_btree), P n1 -> P (NNode n n0 n1)) ->
adamc@29 431 forall n : nat_btree, P n
adam@302 432 ]]
adam@315 433
adam@315 434 We have the usual two cases, one for each constructor of [nat_btree]. *)
adamc@30 435
adamc@30 436
adamc@30 437 (** * Parameterized Types *)
adamc@30 438
adam@316 439 (** We can also define %\index{polymorphism}%polymorphic inductive types, as with algebraic datatypes in Haskell and ML.%\index{Gallina terms!list}\index{Gallina terms!Nil}\index{Gallina terms!Cons}\index{Gallina terms!length}\index{Gallina terms!app}% *)
adamc@30 440
adamc@30 441 Inductive list (T : Set) : Set :=
adamc@30 442 | Nil : list T
adamc@30 443 | Cons : T -> list T -> list T.
adamc@30 444
adamc@30 445 Fixpoint length T (ls : list T) : nat :=
adamc@30 446 match ls with
adamc@30 447 | Nil => O
adamc@30 448 | Cons _ ls' => S (length ls')
adamc@30 449 end.
adamc@30 450
adamc@208 451 Fixpoint app T (ls1 ls2 : list T) : list T :=
adamc@30 452 match ls1 with
adamc@30 453 | Nil => ls2
adamc@30 454 | Cons x ls1' => Cons x (app ls1' ls2)
adamc@30 455 end.
adamc@30 456
adamc@30 457 Theorem length_app : forall T (ls1 ls2 : list T), length (app ls1 ls2)
adamc@30 458 = plus (length ls1) (length ls2).
adamc@41 459 (* begin thide *)
adamc@30 460 induction ls1; crush.
adamc@30 461 Qed.
adamc@41 462 (* end thide *)
adamc@30 463
adam@398 464 (** There is a useful shorthand for writing many definitions that share the same parameter, based on Coq's %\index{sections}\index{Vernacular commands!Section}\index{Vernacular commands!Variable}%_section_ mechanism. The following block of code is equivalent to the above: *)
adamc@30 465
adamc@30 466 (* begin hide *)
adamc@30 467 Reset list.
adamc@30 468 (* end hide *)
adamc@30 469
adamc@30 470 Section list.
adamc@30 471 Variable T : Set.
adamc@30 472
adamc@30 473 Inductive list : Set :=
adamc@30 474 | Nil : list
adamc@30 475 | Cons : T -> list -> list.
adamc@30 476
adamc@30 477 Fixpoint length (ls : list) : nat :=
adamc@30 478 match ls with
adamc@30 479 | Nil => O
adamc@30 480 | Cons _ ls' => S (length ls')
adamc@30 481 end.
adamc@30 482
adamc@208 483 Fixpoint app (ls1 ls2 : list) : list :=
adamc@30 484 match ls1 with
adamc@30 485 | Nil => ls2
adamc@30 486 | Cons x ls1' => Cons x (app ls1' ls2)
adamc@30 487 end.
adamc@30 488
adamc@30 489 Theorem length_app : forall ls1 ls2 : list, length (app ls1 ls2)
adamc@30 490 = plus (length ls1) (length ls2).
adamc@41 491 (* begin thide *)
adamc@30 492 induction ls1; crush.
adamc@30 493 Qed.
adamc@41 494 (* end thide *)
adamc@30 495 End list.
adamc@30 496
adamc@35 497 (* begin hide *)
adamc@35 498 Implicit Arguments Nil [T].
adamc@35 499 (* end hide *)
adamc@35 500
adamc@210 501 (** 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@30 502
adamc@202 503 Print list.
adamc@208 504 (** %\vspace{-.15in}% [[
adamc@208 505 Inductive list (T : Set) : Set :=
adam@316 506 Nil : list T | Cons : T -> list T -> list T
adamc@208 507
adamc@202 508 ]]
adamc@30 509
adamc@202 510 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@30 511
adamc@30 512 Check length.
adamc@208 513 (** %\vspace{-.15in}% [[
adamc@208 514 length
adamc@30 515 : forall T : Set, list T -> nat
adamc@30 516 ]]
adamc@30 517
adamc@202 518 The parameter [T] is treated as a new argument to the induction principle, too. *)
adamc@30 519
adamc@30 520 Check list_ind.
adamc@208 521 (** %\vspace{-.15in}% [[
adamc@208 522 list_ind
adamc@30 523 : forall (T : Set) (P : list T -> Prop),
adamc@30 524 P (Nil T) ->
adamc@30 525 (forall (t : T) (l : list T), P l -> P (Cons t l)) ->
adamc@30 526 forall l : list T, P l
adamc@30 527 ]]
adamc@30 528
adamc@30 529 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@31 530
adamc@31 531
adamc@31 532 (** * Mutually Inductive Types *)
adamc@31 533
adamc@31 534 (** We can define inductive types that refer to each other: *)
adamc@31 535
adamc@31 536 Inductive even_list : Set :=
adamc@31 537 | ENil : even_list
adamc@31 538 | ECons : nat -> odd_list -> even_list
adamc@31 539
adamc@31 540 with odd_list : Set :=
adamc@31 541 | OCons : nat -> even_list -> odd_list.
adamc@31 542
adamc@31 543 Fixpoint elength (el : even_list) : nat :=
adamc@31 544 match el with
adamc@31 545 | ENil => O
adamc@31 546 | ECons _ ol => S (olength ol)
adamc@31 547 end
adamc@31 548
adamc@31 549 with olength (ol : odd_list) : nat :=
adamc@31 550 match ol with
adamc@31 551 | OCons _ el => S (elength el)
adamc@31 552 end.
adamc@31 553
adamc@208 554 Fixpoint eapp (el1 el2 : even_list) : even_list :=
adamc@31 555 match el1 with
adamc@31 556 | ENil => el2
adamc@31 557 | ECons n ol => ECons n (oapp ol el2)
adamc@31 558 end
adamc@31 559
adamc@208 560 with oapp (ol : odd_list) (el : even_list) : odd_list :=
adamc@31 561 match ol with
adamc@31 562 | OCons n el' => OCons n (eapp el' el)
adamc@31 563 end.
adamc@31 564
adamc@31 565 (** Everything is going roughly the same as in past examples, until we try to prove a theorem similar to those that came before. *)
adamc@31 566
adamc@31 567 Theorem elength_eapp : forall el1 el2 : even_list,
adamc@31 568 elength (eapp el1 el2) = plus (elength el1) (elength el2).
adamc@41 569 (* begin thide *)
adamc@31 570 induction el1; crush.
adamc@31 571
adamc@31 572 (** One goal remains: [[
adamc@31 573
adamc@31 574 n : nat
adamc@31 575 o : odd_list
adamc@31 576 el2 : even_list
adamc@31 577 ============================
adamc@31 578 S (olength (oapp o el2)) = S (plus (olength o) (elength el2))
adamc@31 579 ]]
adamc@31 580
adamc@31 581 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@31 582
adamc@31 583 Abort.
adamc@31 584 Check even_list_ind.
adamc@208 585 (** %\vspace{-.15in}% [[
adamc@208 586 even_list_ind
adamc@31 587 : forall P : even_list -> Prop,
adamc@31 588 P ENil ->
adamc@31 589 (forall (n : nat) (o : odd_list), P (ECons n o)) ->
adamc@31 590 forall e : even_list, P e
adamc@208 591
adamc@31 592 ]]
adamc@31 593
adam@316 594 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 %\index{Vernacular commands!Scheme}%[Scheme] command. *)
adamc@31 595
adamc@31 596 Scheme even_list_mut := Induction for even_list Sort Prop
adamc@31 597 with odd_list_mut := Induction for odd_list Sort Prop.
adamc@31 598
adam@316 599 (** This invocation of [Scheme] asks for the creation of induction principles [even_list_mut] for the type [even_list] and [odd_list_mut] for the type [odd_list]. The [Induction] keyword says we want standard induction schemes, since [Scheme] supports more exotic choices. Finally, [Sort Prop] establishes that we really want induction schemes, not recursion schemes, which are the same according to Curry-Howard, save for the [Prop]/[Set] distinction. *)
adam@316 600
adamc@31 601 Check even_list_mut.
adamc@208 602 (** %\vspace{-.15in}% [[
adamc@208 603 even_list_mut
adamc@31 604 : forall (P : even_list -> Prop) (P0 : odd_list -> Prop),
adamc@31 605 P ENil ->
adamc@31 606 (forall (n : nat) (o : odd_list), P0 o -> P (ECons n o)) ->
adamc@31 607 (forall (n : nat) (e : even_list), P e -> P0 (OCons n e)) ->
adamc@31 608 forall e : even_list, P e
adamc@208 609
adamc@31 610 ]]
adamc@31 611
adam@316 612 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.%\index{tactics!apply}% *)
adamc@31 613
adamc@31 614 Theorem n_plus_O' : forall n : nat, plus n O = n.
adamc@31 615 apply (nat_ind (fun n => plus n O = n)); crush.
adamc@31 616 Qed.
adamc@31 617
adamc@31 618 (** 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@31 619
adamc@31 620 This technique generalizes to our mutual example: *)
adamc@31 621
adamc@31 622 Theorem elength_eapp : forall el1 el2 : even_list,
adamc@31 623 elength (eapp el1 el2) = plus (elength el1) (elength el2).
adamc@41 624
adamc@31 625 apply (even_list_mut
adamc@31 626 (fun el1 : even_list => forall el2 : even_list,
adamc@31 627 elength (eapp el1 el2) = plus (elength el1) (elength el2))
adamc@31 628 (fun ol : odd_list => forall el : even_list,
adamc@31 629 olength (oapp ol el) = plus (olength ol) (elength el))); crush.
adamc@31 630 Qed.
adamc@41 631 (* end thide *)
adamc@31 632
adamc@31 633 (** 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@33 634
adamc@33 635
adamc@33 636 (** * Reflexive Types *)
adamc@33 637
adam@398 638 (** A kind of inductive type called a _reflexive type_ 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. Our example will be an encoding of the syntax of first-order logic. Since the idea of syntactic encodings of logic may require a bit of acclimation, let us first consider a simpler formula type for a subset of propositional logic. *)
adam@316 639
adam@316 640 Inductive pformula : Set :=
adam@316 641 | Truth : pformula
adam@316 642 | Falsehood : pformula
adam@316 643 | Conjunction : pformula -> pformula -> pformula.
adam@316 644
adam@398 645 (** A key distinction here is between, for instance, the _syntax_ [Truth] and its _semantics_ [True]. We can make the semantics explicit with a recursive function. This function uses the infix operator %\index{Gallina operators!/\textbackslash}%[/\], which desugars to uses of the type family %\index{Gallina terms!and}%[and] from the standard library. The family [and] implements conjunction, the [Prop] Curry-Howard analogue of the usual pair type from functional programming (which is the type family %\index{Gallina terms!prod}%[prod] in Coq's standard library). *)
adam@316 646
adam@316 647 Fixpoint pformulaDenote (f : pformula) : Prop :=
adam@316 648 match f with
adam@316 649 | Truth => True
adam@316 650 | Falsehood => False
adam@316 651 | Conjunction f1 f2 => pformulaDenote f1 /\ pformulaDenote f2
adam@316 652 end.
adam@316 653
adam@392 654 (** This is just a warm-up that does not use reflexive types, the new feature we mean to introduce. When we set our sights on first-order logic instead, it becomes very handy to give constructors recursive arguments that are functions. *)
adamc@33 655
adamc@33 656 Inductive formula : Set :=
adamc@33 657 | Eq : nat -> nat -> formula
adamc@33 658 | And : formula -> formula -> formula
adamc@33 659 | Forall : (nat -> formula) -> formula.
adamc@33 660
adam@316 661 (** 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]:%\index{Vernacular commands!Example}% *)
adamc@33 662
adamc@33 663 Example forall_refl : formula := Forall (fun x => Eq x x).
adamc@33 664
adamc@33 665 (** We can write recursive functions over reflexive types quite naturally. Here is one translating our formulas into native Coq propositions. *)
adamc@33 666
adamc@33 667 Fixpoint formulaDenote (f : formula) : Prop :=
adamc@33 668 match f with
adamc@33 669 | Eq n1 n2 => n1 = n2
adamc@33 670 | And f1 f2 => formulaDenote f1 /\ formulaDenote f2
adamc@33 671 | Forall f' => forall n : nat, formulaDenote (f' n)
adamc@33 672 end.
adamc@33 673
adamc@33 674 (** We can also encode a trivial formula transformation that swaps the order of equality and conjunction operands. *)
adamc@33 675
adamc@33 676 Fixpoint swapper (f : formula) : formula :=
adamc@33 677 match f with
adamc@33 678 | Eq n1 n2 => Eq n2 n1
adamc@33 679 | And f1 f2 => And (swapper f2) (swapper f1)
adamc@33 680 | Forall f' => Forall (fun n => swapper (f' n))
adamc@33 681 end.
adamc@33 682
adamc@33 683 (** It is helpful to prove that this transformation does not make true formulas false. *)
adamc@33 684
adamc@33 685 Theorem swapper_preserves_truth : forall f, formulaDenote f -> formulaDenote (swapper f).
adamc@41 686 (* begin thide *)
adamc@33 687 induction f; crush.
adamc@33 688 Qed.
adamc@41 689 (* end thide *)
adamc@33 690
adamc@33 691 (** We can take a look at the induction principle behind this proof. *)
adamc@33 692
adamc@33 693 Check formula_ind.
adamc@208 694 (** %\vspace{-.15in}% [[
adamc@208 695 formula_ind
adamc@33 696 : forall P : formula -> Prop,
adamc@33 697 (forall n n0 : nat, P (Eq n n0)) ->
adamc@33 698 (forall f0 : formula,
adamc@33 699 P f0 -> forall f1 : formula, P f1 -> P (And f0 f1)) ->
adamc@33 700 (forall f1 : nat -> formula,
adamc@33 701 (forall n : nat, P (f1 n)) -> P (Forall f1)) ->
adamc@33 702 forall f2 : formula, P f2
adamc@208 703
adamc@208 704 ]]
adamc@33 705
adam@398 706 Focusing on the [Forall] case, which comes third, we see that we are allowed to assume that the theorem holds _for any application of the argument function [f1]_. 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@33 707
adamc@33 708 %\medskip%
adamc@33 709
adam@316 710 Up to this point, we have seen how to encode in Coq more and more of what is possible with algebraic datatypes in %\index{Haskell}%Haskell and %\index{ML}%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@33 711
adam@400 712 Given our last example of an inductive type, many readers are probably eager to try encoding the syntax of %\index{lambda calculus}%lambda calculus. Indeed, the function-based representation technique that we just used, called%\index{higher-order abstract syntax}\index{HOAS|see{higher-order abstract syntax}}% _higher-order abstract syntax_ (HOAS)%~\cite{HOAS}%, is the representation of choice for lambda calculi in %\index{Twelf}%Twelf and in many applications implemented in Haskell and ML. Let us try to import that choice to Coq: *)
adam@400 713 (* begin hide *)
adam@400 714 Inductive term : Set := App | Abs.
adam@400 715 Reset term.
adam@400 716 (* end hide *)
adamc@33 717 (** [[
adamc@33 718 Inductive term : Set :=
adamc@33 719 | App : term -> term -> term
adamc@33 720 | Abs : (term -> term) -> term.
adamc@33 721 ]]
adamc@33 722
adam@316 723 <<
adam@316 724 Error: Non strictly positive occurrence of "term" in "(term -> term) -> term"
adam@316 725 >>
adam@316 726
adam@400 727 We have run afoul of the%\index{strict positivity requirement}\index{positivity requirement}% _strict positivity requirement_ 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. Our candidate definition above violates the positivity requirement because it involves an argument of type [term -> term], where the type [term] that we are defining appears to the left of an arrow. The candidate type of [App] is fine, however, since every occurrence of [term] is either a constructor argument or the final result type.
adamc@33 728
adamc@33 729 Why must Coq enforce this restriction? Imagine that our last definition had been accepted, allowing us to write this function:
adamc@33 730
adamc@33 731 [[
adamc@33 732 Definition uhoh (t : term) : term :=
adamc@33 733 match t with
adamc@33 734 | Abs f => f t
adamc@33 735 | _ => t
adamc@33 736 end.
adamc@33 737
adamc@205 738 ]]
adamc@205 739
adamc@33 740 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@33 741
adam@316 742 %\index{termination checking}%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@33 743
adamc@33 744 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@34 745
adamc@34 746
adam@317 747 (** * An Interlude on Induction Principles *)
adamc@34 748
adam@317 749 (** 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 %\index{induction principles}%induction principles we have used. A close look at the details here will help us construct induction principles manually, which we will see is necessary for some more advanced inductive definitions. *)
adamc@34 750
adam@317 751 (* begin hide *)
adamc@34 752 Print unit_ind.
adam@317 753 (* end hide *)
adam@317 754 (** %\noindent%[Print] [unit_ind.] *)
adam@317 755 (** [[
adamc@208 756 unit_ind =
adamc@208 757 fun P : unit -> Prop => unit_rect P
adamc@34 758 : forall P : unit -> Prop, P tt -> forall u : unit, P u
adamc@208 759
adamc@34 760 ]]
adamc@34 761
adam@317 762 We see that this induction principle is defined in terms of a more general principle, [unit_rect]. The %\texttt{%#<tt>#rec#</tt>#%}% stands for %``%#"#recursion principle,#"#%''% and the %\texttt{%#<tt>#t#</tt>#%}% at the end stands for [Type]. *)
adamc@34 763
adamc@34 764 Check unit_rect.
adamc@208 765 (** %\vspace{-.15in}% [[
adamc@208 766 unit_rect
adamc@34 767 : forall P : unit -> Type, P tt -> forall u : unit, P u
adamc@208 768
adamc@34 769 ]]
adamc@34 770
adam@400 771 The principle [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@34 772
adam@317 773 (* begin hide *)
adamc@34 774 Print unit_rec.
adam@317 775 (* end hide *)
adam@317 776 (** %\noindent%[Print] [unit_rec.] *)
adam@317 777 (** [[
adamc@208 778 unit_rec =
adamc@208 779 fun P : unit -> Set => unit_rect P
adamc@34 780 : forall P : unit -> Set, P tt -> forall u : unit, P u
adamc@208 781
adamc@34 782 ]]
adamc@34 783
adam@317 784 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 %\index{recursion principles}%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@34 785
adamc@34 786 Definition always_O (u : unit) : nat :=
adamc@34 787 match u with
adamc@34 788 | tt => O
adamc@34 789 end.
adamc@34 790
adamc@34 791 Definition always_O' (u : unit) : nat :=
adamc@34 792 unit_rec (fun _ : unit => nat) O u.
adamc@34 793
adamc@34 794 (** 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@34 795
adam@317 796 (* begin hide *)
adamc@34 797 Print unit_rect.
adam@317 798 (* end hide *)
adam@317 799 (** %\noindent%[Print] [unit_rect.] *)
adam@317 800 (** [[
adamc@208 801 unit_rect =
adamc@208 802 fun (P : unit -> Type) (f : P tt) (u : unit) =>
adamc@208 803 match u as u0 return (P u0) with
adamc@208 804 | tt => f
adamc@208 805 end
adamc@34 806 : forall P : unit -> Type, P tt -> forall u : unit, P u
adamc@208 807
adamc@34 808 ]]
adamc@34 809
adam@400 810 The only new wrinkle here is the annotations on the [match] expression. This is a%\index{dependent pattern matching}% _dependently typed_ pattern match, because the _type_ of the expression depends on the _value_ being matched on. Of course, for this example, the dependency is degenerate; the value being matched on has type [unit], so it may only take on a single known value, [tt]. We will meet more involved examples later, especially in Part II of the book.
adam@317 811
adam@317 812 %\index{type inference}%Type inference for dependent pattern matching is undecidable, which can be proved by reduction from %\index{higher-order unification}%higher-order unification%~\cite{HOU}%. Thus, we often find ourselves needing to annotate our programs in a way that explains dependencies to the type checker. In the example of [unit_rect], we have an %\index{Gallina terms!as}%[as] clause, which binds a name for the discriminee; and a %\index{Gallina terms!return}%[return] clause, which gives a way to compute the [match] result type as a function of the discriminee.
adamc@34 813
adamc@34 814 To prove that [unit_rect] is nothing special, we can reimplement it manually. *)
adamc@34 815
adamc@34 816 Definition unit_rect' (P : unit -> Type) (f : P tt) (u : unit) :=
adamc@208 817 match u with
adamc@34 818 | tt => f
adamc@34 819 end.
adamc@34 820
adam@317 821 (** We rely on Coq's heuristics for inferring [match] annotations, which are not consulted in the pretty-printing of terms.
adamc@34 822
adam@400 823 We can check the implementation [nat_rect] as well: *)
adamc@34 824
adamc@34 825 Print nat_rect.
adam@317 826
adam@317 827 (** %\vspace{-.05in}% [[
adam@400 828 nat_rect =
adamc@208 829 fun (P : nat -> Type) (f : P O) (f0 : forall n : nat, P n -> P (S n)) =>
adamc@208 830 fix F (n : nat) : P n :=
adamc@208 831 match n as n0 return (P n0) with
adamc@208 832 | O => f
adamc@208 833 | S n0 => f0 n0 (F n0)
adamc@208 834 end
adamc@208 835 : forall P : nat -> Type,
adamc@208 836 P O -> (forall n : nat, P n -> P (S n)) -> forall n : nat, P n
adamc@208 837 ]]
adamc@34 838
adam@400 839 Now we have an actual recursive definition. Expressions starting with %\index{Gallina terms!fix}%[fix] are anonymous forms 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@34 840
adam@317 841 Section nat_ind'.
adamc@208 842 (** First, we have the property of natural numbers that we aim to prove. *)
adamc@34 843
adam@317 844 Variable P : nat -> Prop.
adamc@34 845
adam@317 846 (** Then we require a proof of the [O] case, which we declare with the command %\index{Vernacular commands!Hypothesis}%[Hypothesis], which is a synonym for [Variable] that, by convention, is used for variables whose types are propositions. *)
adamc@34 847
adam@317 848 Hypothesis O_case : P O.
adamc@34 849
adamc@208 850 (** Next is a proof of the [S] case, which may assume an inductive hypothesis. *)
adamc@34 851
adam@317 852 Hypothesis S_case : forall n : nat, P n -> P (S n).
adamc@34 853
adamc@208 854 (** Finally, we define a recursive function to tie the pieces together. *)
adamc@34 855
adam@317 856 Fixpoint nat_ind' (n : nat) : P n :=
adam@317 857 match n with
adam@317 858 | O => O_case
adam@317 859 | S n' => S_case (nat_ind' n')
adam@317 860 end.
adam@317 861 End nat_ind'.
adamc@34 862
adam@400 863 (** 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@34 864
adam@317 865 %\medskip%
adamc@34 866
adam@317 867 We can also examine the definition of [even_list_mut], which we generated with [Scheme] for a mutually recursive type. *)
adamc@34 868
adam@317 869 Print even_list_mut.
adam@317 870 (** [[
adam@317 871 even_list_mut =
adam@317 872 fun (P : even_list -> Prop) (P0 : odd_list -> Prop)
adam@317 873 (f : P ENil) (f0 : forall (n : nat) (o : odd_list), P0 o -> P (ECons n o))
adam@317 874 (f1 : forall (n : nat) (e : even_list), P e -> P0 (OCons n e)) =>
adam@317 875 fix F (e : even_list) : P e :=
adam@317 876 match e as e0 return (P e0) with
adam@317 877 | ENil => f
adam@317 878 | ECons n o => f0 n o (F0 o)
adam@317 879 end
adam@317 880 with F0 (o : odd_list) : P0 o :=
adam@317 881 match o as o0 return (P0 o0) with
adam@317 882 | OCons n e => f1 n e (F e)
adam@317 883 end
adam@317 884 for F
adam@317 885 : forall (P : even_list -> Prop) (P0 : odd_list -> Prop),
adam@317 886 P ENil ->
adam@317 887 (forall (n : nat) (o : odd_list), P0 o -> P (ECons n o)) ->
adam@317 888 (forall (n : nat) (e : even_list), P e -> P0 (OCons n e)) ->
adam@317 889 forall e : even_list, P e
adamc@34 890
adam@317 891 ]]
adamc@34 892
adam@317 893 We see a mutually recursive [fix], with the different functions separated by %\index{Gallina terms!with}%[with] in the same way that they would be separated by %\texttt{%#<tt>#and#</tt>#%}% in ML. A final %\index{Gallina terms!for}%[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@208 894
adam@317 895 Section even_list_mut'.
adam@317 896 (** First, we need the properties that we are proving. *)
adamc@208 897
adam@317 898 Variable Peven : even_list -> Prop.
adam@317 899 Variable Podd : odd_list -> Prop.
adamc@208 900
adam@317 901 (** Next, we need proofs of the three cases. *)
adamc@208 902
adam@317 903 Hypothesis ENil_case : Peven ENil.
adam@317 904 Hypothesis ECons_case : forall (n : nat) (o : odd_list), Podd o -> Peven (ECons n o).
adam@317 905 Hypothesis OCons_case : forall (n : nat) (e : even_list), Peven e -> Podd (OCons n e).
adamc@208 906
adam@317 907 (** Finally, we define the recursive functions. *)
adamc@208 908
adam@317 909 Fixpoint even_list_mut' (e : even_list) : Peven e :=
adam@317 910 match e with
adam@317 911 | ENil => ENil_case
adam@317 912 | ECons n o => ECons_case n (odd_list_mut' o)
adam@317 913 end
adam@317 914 with odd_list_mut' (o : odd_list) : Podd o :=
adam@317 915 match o with
adam@317 916 | OCons n e => OCons_case n (even_list_mut' e)
adam@317 917 end.
adamc@34 918 End even_list_mut'.
adamc@34 919
adamc@34 920 (** 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@34 921
adamc@34 922 Section formula_ind'.
adamc@34 923 Variable P : formula -> Prop.
adamc@38 924 Hypothesis Eq_case : forall n1 n2 : nat, P (Eq n1 n2).
adamc@38 925 Hypothesis And_case : forall f1 f2 : formula,
adamc@34 926 P f1 -> P f2 -> P (And f1 f2).
adamc@38 927 Hypothesis Forall_case : forall f : nat -> formula,
adamc@34 928 (forall n : nat, P (f n)) -> P (Forall f).
adamc@34 929
adamc@34 930 Fixpoint formula_ind' (f : formula) : P f :=
adamc@208 931 match f with
adamc@34 932 | Eq n1 n2 => Eq_case n1 n2
adamc@34 933 | And f1 f2 => And_case (formula_ind' f1) (formula_ind' f2)
adamc@34 934 | Forall f' => Forall_case f' (fun n => formula_ind' (f' n))
adamc@34 935 end.
adamc@34 936 End formula_ind'.
adamc@34 937
adam@317 938 (** It is apparent that induction principle implementations involve some tedium but not terribly much creativity. *)
adam@317 939
adamc@35 940
adamc@35 941 (** * Nested Inductive Types *)
adamc@35 942
adamc@35 943 (** 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@35 944
adamc@35 945 Inductive nat_tree : Set :=
adamc@35 946 | NLeaf' : nat_tree
adamc@35 947 | NNode' : nat -> list nat_tree -> nat_tree.
adamc@35 948
adam@398 949 (** This is an example of a %\index{nested inductive type}%_nested_ 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@35 950
adam@317 951 Like we encountered for mutual inductive types, we find that the automatically generated induction principle for [nat_tree] is too weak. *)
adamc@35 952
adamc@35 953 Check nat_tree_ind.
adamc@208 954 (** %\vspace{-.15in}% [[
adamc@208 955 nat_tree_ind
adamc@35 956 : forall P : nat_tree -> Prop,
adamc@35 957 P NLeaf' ->
adamc@35 958 (forall (n : nat) (l : list nat_tree), P (NNode' n l)) ->
adamc@35 959 forall n : nat_tree, P n
adamc@208 960
adamc@35 961 ]]
adamc@35 962
adam@317 963 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 of different type families. This is roughly the same creativity employed in the traditional task of strengthening an induction hypothesis. Now that we know how to implement induction principles manually, we are in a position to apply just such creativity to this problem.
adamc@35 964
adamc@35 965 First, we will need an auxiliary definition, characterizing what it means for a property to hold of every element of a list. *)
adamc@35 966
adamc@35 967 Section All.
adamc@35 968 Variable T : Set.
adamc@35 969 Variable P : T -> Prop.
adamc@35 970
adamc@35 971 Fixpoint All (ls : list T) : Prop :=
adamc@35 972 match ls with
adamc@35 973 | Nil => True
adamc@35 974 | Cons h t => P h /\ All t
adamc@35 975 end.
adamc@35 976 End All.
adamc@35 977
adam@317 978 (** It will be useful to review the definitions of [True] and [/\], since we will want to write manual proofs of them below. *)
adamc@35 979
adamc@35 980 Print True.
adam@317 981 (** [[
adamc@208 982 Inductive True : Prop := I : True
adamc@208 983 ]]
adamc@35 984
adamc@35 985 That is, [True] is a proposition with exactly one proof, [I], which we may always supply trivially.
adamc@35 986
adam@400 987 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 %\index{Vernacular commands!Locate}%[Locate] command, whose argument may be a parsing token.%\index{Gallina terms!and}% *)
adamc@35 988
adamc@35 989 Locate "/\".
adam@317 990 (** [[
adam@317 991 "A /\ B" := and A B : type_scope (default interpretation)
adam@302 992 ]]
adam@302 993 *)
adamc@35 994
adamc@35 995 Print and.
adam@317 996 (** [[
adamc@208 997 Inductive and (A : Prop) (B : Prop) : Prop := conj : A -> B -> A /\ B
adam@317 998 ]]
adam@317 999 %\vspace{-.1in}%
adam@317 1000 <<
adamc@208 1001 For conj: Arguments A, B are implicit
adam@317 1002 >>
adamc@35 1003
adam@400 1004 In addition to the definition of [and] itself, we get information on %\index{implicit arguments}%implicit arguments (and some other information that we omit here). 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 1005
adamc@35 1006 %\medskip%
adamc@35 1007
adamc@35 1008 Now we create a section for our induction principle, following the same basic plan as in the last section of this chapter. *)
adamc@35 1009
adamc@35 1010 Section nat_tree_ind'.
adamc@35 1011 Variable P : nat_tree -> Prop.
adamc@35 1012
adamc@38 1013 Hypothesis NLeaf'_case : P NLeaf'.
adamc@38 1014 Hypothesis NNode'_case : forall (n : nat) (ls : list nat_tree),
adamc@35 1015 All P ls -> P (NNode' n ls).
adamc@35 1016
adamc@35 1017 (** 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 1018
adamc@35 1019 [[
adamc@35 1020 Fixpoint nat_tree_ind' (tr : nat_tree) : P tr :=
adamc@208 1021 match tr with
adamc@35 1022 | NLeaf' => NLeaf'_case
adamc@35 1023 | NNode' n ls => NNode'_case n ls (list_nat_tree_ind ls)
adamc@35 1024 end
adamc@35 1025
adamc@35 1026 with list_nat_tree_ind (ls : list nat_tree) : All P ls :=
adamc@208 1027 match ls with
adamc@35 1028 | Nil => I
adamc@35 1029 | Cons tr rest => conj (nat_tree_ind' tr) (list_nat_tree_ind rest)
adamc@35 1030 end.
adamc@35 1031
adamc@205 1032 ]]
adamc@205 1033
adam@317 1034 Coq rejects this definition, saying
adam@317 1035 <<
adam@317 1036 Recursive call to nat_tree_ind' has principal argument equal to "tr"
adam@317 1037 instead of rest.
adam@317 1038 >>
adam@317 1039
adam@317 1040 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 1041
adamc@35 1042 Fixpoint nat_tree_ind' (tr : nat_tree) : P tr :=
adamc@208 1043 match tr with
adamc@35 1044 | NLeaf' => NLeaf'_case
adamc@35 1045 | NNode' n ls => NNode'_case n ls
adamc@35 1046 ((fix list_nat_tree_ind (ls : list nat_tree) : All P ls :=
adamc@208 1047 match ls with
adamc@35 1048 | Nil => I
adamc@35 1049 | Cons tr rest => conj (nat_tree_ind' tr) (list_nat_tree_ind rest)
adamc@35 1050 end) ls)
adamc@35 1051 end.
adamc@35 1052
adam@398 1053 (** We include an anonymous [fix] version of [list_nat_tree_ind] that is literally _nested_ inside the definition of the recursive function corresponding to the inductive definition that had the nested use of [list]. *)
adamc@35 1054
adamc@35 1055 End nat_tree_ind'.
adamc@35 1056
adamc@35 1057 (** 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 1058
adamc@35 1059 Section map.
adamc@35 1060 Variables T T' : Set.
adam@317 1061 Variable F : T -> T'.
adamc@35 1062
adamc@35 1063 Fixpoint map (ls : list T) : list T' :=
adamc@35 1064 match ls with
adamc@35 1065 | Nil => Nil
adam@317 1066 | Cons h t => Cons (F h) (map t)
adamc@35 1067 end.
adamc@35 1068 End map.
adamc@35 1069
adamc@35 1070 Fixpoint sum (ls : list nat) : nat :=
adamc@35 1071 match ls with
adamc@35 1072 | Nil => O
adamc@35 1073 | Cons h t => plus h (sum t)
adamc@35 1074 end.
adamc@35 1075
adamc@35 1076 (** Now we can define a size function over our trees. *)
adamc@35 1077
adamc@35 1078 Fixpoint ntsize (tr : nat_tree) : nat :=
adamc@35 1079 match tr with
adamc@35 1080 | NLeaf' => S O
adamc@35 1081 | NNode' _ trs => S (sum (map ntsize trs))
adamc@35 1082 end.
adamc@35 1083
adamc@35 1084 (** 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 1085
adamc@208 1086 Fixpoint ntsplice (tr1 tr2 : nat_tree) : nat_tree :=
adamc@35 1087 match tr1 with
adamc@35 1088 | NLeaf' => NNode' O (Cons tr2 Nil)
adamc@35 1089 | NNode' n Nil => NNode' n (Cons tr2 Nil)
adamc@35 1090 | NNode' n (Cons tr trs) => NNode' n (Cons (ntsplice tr tr2) trs)
adamc@35 1091 end.
adamc@35 1092
adamc@35 1093 (** 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 1094
adamc@41 1095 (* begin thide *)
adamc@35 1096 Lemma plus_S : forall n1 n2 : nat,
adamc@35 1097 plus n1 (S n2) = S (plus n1 n2).
adamc@35 1098 induction n1; crush.
adamc@35 1099 Qed.
adamc@41 1100 (* end thide *)
adamc@35 1101
adamc@35 1102 (** Now we begin the proof of the theorem, adding the lemma [plus_S] as a hint. *)
adamc@35 1103
adamc@35 1104 Theorem ntsize_ntsplice : forall tr1 tr2 : nat_tree, ntsize (ntsplice tr1 tr2)
adamc@35 1105 = plus (ntsize tr2) (ntsize tr1).
adamc@41 1106 (* begin thide *)
adam@375 1107 Hint Rewrite plus_S.
adamc@35 1108
adam@317 1109 (** We know that the standard induction principle is insufficient for the task, so we need to provide a %\index{tactics!using}%[using] clause for the [induction] tactic to specify our alternate principle. *)
adamc@208 1110
adamc@35 1111 induction tr1 using nat_tree_ind'; crush.
adamc@35 1112
adamc@35 1113 (** One subgoal remains: [[
adamc@35 1114 n : nat
adamc@35 1115 ls : list nat_tree
adamc@35 1116 H : All
adamc@35 1117 (fun tr1 : nat_tree =>
adamc@35 1118 forall tr2 : nat_tree,
adamc@35 1119 ntsize (ntsplice tr1 tr2) = plus (ntsize tr2) (ntsize tr1)) ls
adamc@35 1120 tr2 : nat_tree
adamc@35 1121 ============================
adamc@35 1122 ntsize
adamc@35 1123 match ls with
adamc@35 1124 | Nil => NNode' n (Cons tr2 Nil)
adamc@35 1125 | Cons tr trs => NNode' n (Cons (ntsplice tr tr2) trs)
adamc@35 1126 end = S (plus (ntsize tr2) (sum (map ntsize ls)))
adamc@208 1127
adamc@35 1128 ]]
adamc@35 1129
adamc@35 1130 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 1131
adamc@35 1132 destruct ls; crush.
adamc@35 1133
adam@317 1134 (** We can go further in automating the proof by exploiting the hint mechanism.%\index{Vernacular commands!Hint Extern}% *)
adamc@35 1135
adamc@35 1136 Restart.
adam@317 1137
adamc@35 1138 Hint Extern 1 (ntsize (match ?LS with Nil => _ | Cons _ _ => _ end) = _) =>
adamc@35 1139 destruct LS; crush.
adamc@35 1140 induction tr1 using nat_tree_ind'; crush.
adamc@35 1141 Qed.
adamc@41 1142 (* end thide *)
adamc@35 1143
adamc@35 1144 (** 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 1145
adam@317 1146 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, and the hint can continue working even if the rest of the proof structure changes significantly. *)
adamc@36 1147
adamc@36 1148
adamc@36 1149 (** * Manual Proofs About Constructors *)
adamc@36 1150
adam@317 1151 (** It can be useful to understand how tactics like %\index{tactics!discriminate}%[discriminate] and %\index{tactics!injection}%[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 1152
adamc@36 1153 Theorem true_neq_false : true <> false.
adamc@208 1154
adamc@41 1155 (* begin thide *)
adam@317 1156 (** We begin with the tactic %\index{tactics!red}%[red], which is short for %``%#"#one step of reduction,#"#%''% to unfold the definition of logical negation. *)
adamc@36 1157
adamc@36 1158 red.
adamc@36 1159 (** [[
adamc@36 1160 ============================
adamc@36 1161 true = false -> False
adamc@208 1162
adamc@36 1163 ]]
adamc@36 1164
adam@317 1165 The negation is replaced with an implication of falsehood. We use the tactic %\index{tactics!intro}%[intro H] to change the assumption of the implication into a hypothesis named [H]. *)
adamc@36 1166
adamc@36 1167 intro H.
adamc@36 1168 (** [[
adamc@36 1169 H : true = false
adamc@36 1170 ============================
adamc@36 1171 False
adamc@208 1172
adamc@36 1173 ]]
adamc@36 1174
adamc@36 1175 This is the point in the proof where we apply some creativity. We define a function whose utility will become clear soon. *)
adamc@36 1176
adam@317 1177 Definition toProp (b : bool) := if b then True else False.
adamc@36 1178
adam@400 1179 (** 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 [toProp] such that our conclusion of [False] is computationally equivalent to [toProp false]. Thus, the %\index{tactics!change}%[change] tactic will let us change the conclusion to [toProp false]. The general form [change e] replaces the conclusion with [e], whenever Coq's built-in computation rules suffice to establish the equivalence of [e] with the original conclusion. *)
adamc@36 1180
adam@317 1181 change (toProp false).
adamc@36 1182 (** [[
adamc@36 1183 H : true = false
adamc@36 1184 ============================
adam@317 1185 toProp false
adamc@208 1186
adamc@36 1187 ]]
adamc@36 1188
adam@317 1189 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.%\index{tactics!rewrite}% *)
adamc@36 1190
adamc@36 1191 rewrite <- H.
adamc@36 1192 (** [[
adamc@36 1193 H : true = false
adamc@36 1194 ============================
adam@317 1195 toProp true
adamc@208 1196
adamc@36 1197 ]]
adamc@36 1198
adamc@36 1199 We are almost done. Just how close we are to done is revealed by computational simplification. *)
adamc@36 1200
adamc@36 1201 simpl.
adamc@36 1202 (** [[
adamc@36 1203 H : true = false
adamc@36 1204 ============================
adamc@36 1205 True
adamc@208 1206
adam@302 1207 ]]
adam@302 1208 *)
adamc@36 1209
adamc@36 1210 trivial.
adamc@36 1211 Qed.
adamc@41 1212 (* end thide *)
adamc@36 1213
adamc@36 1214 (** I have no trivial automated version of this proof to suggest, beyond using [discriminate] or [congruence] in the first place.
adamc@36 1215
adamc@36 1216 %\medskip%
adamc@36 1217
adamc@36 1218 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 1219
adamc@36 1220 Theorem S_inj' : forall n m : nat, S n = S m -> n = m.
adamc@41 1221 (* begin thide *)
adamc@36 1222 intros n m H.
adamc@36 1223 change (pred (S n) = pred (S m)).
adamc@36 1224 rewrite H.
adamc@36 1225 reflexivity.
adamc@36 1226 Qed.
adamc@41 1227 (* end thide *)
adamc@36 1228
adam@400 1229 (** The key piece of creativity in this theorem comes in the use of the natural number predecessor function [pred]. Embodied in the implementation of [injection] is a generic recipe for writing such type-specific functions.
adam@317 1230
adam@317 1231 The examples in this section illustrate an important aspect of the design philosophy behind Coq. We could certainly design a Gallina replacement that built in rules for constructor discrimination and injectivity, but a simpler alternative is to include a few carefully chosen rules that enable the desired reasoning patterns and many others. A key benefit of this philosophy is that the complexity of proof checking is minimized, which bolsters our confidence that proved theorems are really true. *)