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author Adam Chlipala <adam@chlipala.net>
date Mon, 30 Jul 2012 16:50:02 -0400
parents 393b8ed99c2f
children 89c67796754e
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@420 30 (** Mainstream presentations of mathematics treat proofs as objects that exist outside of the universe of mathematical objects. However, for a variety of reasoning tasks, 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@439 41 (** %\smallskip{}%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@439 46 (** %\smallskip{}%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@439 53 (** %\smallskip{}%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@439 58 (** %\smallskip{}%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@439 63 (** %\smallskip{}%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@439 68 (** %\smallskip{}%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@419 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 (GADTs), by adding the possibility for type dependency. Even so, it is worth backing up from the examples of the last chapter and going over basic, algebraic datatype uses of inductive datatypes, because the chance to prove things about the values of these types adds new wrinkles beyond usual practice in Haskell and ML.
adamc@26 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
adam@439 92 (** %\smallskip{}%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 destruct x.
adamc@205 115 ]]
adamc@205 116
adam@420 117 %\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@26 118
adam@398 119 What exactly _is_ the %\index{induction principles}%induction principle for [unit]? We can ask Coq: *)
adamc@26 120
adamc@26 121 Check unit_ind.
adamc@208 122 (** [unit_ind : forall P : unit -> Prop, P tt -> forall u : unit, P u] *)
adamc@26 123
adam@439 124 (** %\smallskip{}%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@26 125
adam@315 126 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@26 127
adam@400 128 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@315 129
adam@315 130 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@26 131
adamc@26 132 %\medskip%
adamc@26 133
adam@315 134 We can define an inductive type even simpler than [unit]:%\index{Gallina terms!Empty\_set}% *)
adamc@26 135
adamc@26 136 Inductive Empty_set : Set := .
adamc@26 137
adamc@26 138 (** [Empty_set] has no elements. We can prove fun theorems about it: *)
adamc@26 139
adamc@26 140 Theorem the_sky_is_falling : forall x : Empty_set, 2 + 2 = 5.
adamc@41 141 (* begin thide *)
adamc@26 142 destruct 1.
adamc@26 143 Qed.
adamc@41 144 (* end thide *)
adamc@26 145
adam@317 146 (** 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@26 147
adamc@26 148 We can see the induction principle that made this proof so easy: *)
adamc@26 149
adamc@26 150 Check Empty_set_ind.
adam@400 151 (** [Empty_set_ind : forall (P : Empty_set -> Prop) (e : Empty_set), P e] *)
adamc@26 152
adam@439 153 (** %\smallskip{}%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@26 154
adamc@26 155 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@26 156
adamc@26 157 Definition e2u (e : Empty_set) : unit := match e with end.
adamc@26 158
adam@315 159 (** 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@26 160
adamc@26 161 %\medskip%
adamc@26 162
adam@315 163 Moving up the ladder of complexity, we can define the booleans:%\index{Gallina terms!bool}\index{Gallina terms!true}\index{Gallina terms!false}% *)
adamc@26 164
adamc@26 165 Inductive bool : Set :=
adamc@26 166 | true
adamc@26 167 | false.
adamc@26 168
adam@316 169 (** We can use less vacuous pattern matching to define boolean negation.%\index{Gallina terms!negb}% *)
adamc@26 170
adam@279 171 Definition negb (b : bool) : bool :=
adamc@26 172 match b with
adamc@26 173 | true => false
adamc@26 174 | false => true
adamc@26 175 end.
adamc@26 176
adam@317 177 (** 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@27 178
adam@279 179 Definition negb' (b : bool) : bool :=
adamc@27 180 if b then false else true.
adamc@27 181
adam@279 182 (** We might want to prove that [negb] is its own inverse operation. *)
adamc@26 183
adam@279 184 Theorem negb_inverse : forall b : bool, negb (negb b) = b.
adamc@41 185 (* begin thide *)
adamc@26 186 destruct b.
adamc@26 187
adamc@208 188 (** After we case-analyze on [b], we are left with one subgoal for each constructor of [bool].
adam@439 189 [[
adam@439 190 2 subgoals
adamc@26 191
adamc@26 192 ============================
adam@279 193 negb (negb true) = true
adam@439 194
adam@439 195 subgoal 2 is
adam@439 196
adam@279 197 negb (negb false) = false
adamc@26 198 ]]
adamc@26 199
adamc@26 200 The first subgoal follows by Coq's rules of computation, so we can dispatch it easily: *)
adamc@26 201
adamc@26 202 reflexivity.
adamc@26 203
adam@315 204 (** Likewise for the second subgoal, so we can restart the proof and give a very compact justification.%\index{Vernacular commands!Restart}% *)
adamc@26 205
adamc@26 206 Restart.
adam@315 207
adamc@26 208 destruct b; reflexivity.
adamc@26 209 Qed.
adamc@41 210 (* end thide *)
adamc@27 211
adam@315 212 (** Another theorem about booleans illustrates another useful tactic.%\index{tactics!discriminate}% *)
adamc@27 213
adam@279 214 Theorem negb_ineq : forall b : bool, negb b <> b.
adamc@41 215 (* begin thide *)
adamc@27 216 destruct b; discriminate.
adamc@27 217 Qed.
adamc@41 218 (* end thide *)
adamc@27 219
adamc@27 220 (** [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@27 221
adamc@27 222 At this point, it is probably not hard to guess what the underlying induction principle for [bool] is. *)
adamc@27 223
adamc@27 224 Check bool_ind.
adamc@208 225 (** [bool_ind : forall P : bool -> Prop, P true -> P false -> forall b : bool, P b] *)
adamc@28 226
adam@439 227 (** %\smallskip{}%That is, to prove that a property describes all [bool]s, prove that it describes both [true] and [false].
adam@315 228
adam@392 229 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@315 230
adamc@28 231
adamc@28 232 (** * Simple Recursive Types *)
adamc@28 233
adam@315 234 (** 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@28 235
adamc@28 236 Inductive nat : Set :=
adamc@28 237 | O : nat
adamc@28 238 | S : nat -> nat.
adamc@28 239
adam@439 240 (** The constructor [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@28 241
adam@316 242 Pattern matching works as we demonstrated in the last chapter:%\index{Gallina terms!pred}% *)
adamc@28 243
adamc@28 244 Definition isZero (n : nat) : bool :=
adamc@28 245 match n with
adamc@28 246 | O => true
adamc@28 247 | S _ => false
adamc@28 248 end.
adamc@28 249
adamc@28 250 Definition pred (n : nat) : nat :=
adamc@28 251 match n with
adamc@28 252 | O => O
adamc@28 253 | S n' => n'
adamc@28 254 end.
adamc@28 255
adamc@28 256 (** We can prove theorems by case analysis: *)
adamc@28 257
adamc@28 258 Theorem S_isZero : forall n : nat, isZero (pred (S (S n))) = false.
adamc@41 259 (* begin thide *)
adamc@28 260 destruct n; reflexivity.
adamc@28 261 Qed.
adamc@41 262 (* end thide *)
adamc@28 263
adam@316 264 (** 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@28 265
adamc@208 266 Fixpoint plus (n m : nat) : nat :=
adamc@28 267 match n with
adamc@28 268 | O => m
adamc@28 269 | S n' => S (plus n' m)
adamc@28 270 end.
adamc@28 271
adamc@208 272 (** Recall that [Fixpoint] is Coq's mechanism for recursive function definitions. Some theorems about [plus] can be proved without induction. *)
adamc@28 273
adamc@28 274 Theorem O_plus_n : forall n : nat, plus O n = n.
adamc@41 275 (* begin thide *)
adamc@28 276 intro; reflexivity.
adamc@28 277 Qed.
adamc@41 278 (* end thide *)
adamc@28 279
adamc@208 280 (** 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@28 281
adamc@28 282 Theorem n_plus_O : forall n : nat, plus n O = n.
adamc@41 283 (* begin thide *)
adamc@28 284 induction n.
adamc@28 285
adam@398 286 (** Our first subgoal is [plus O O = O], which _is_ trivial by computation. *)
adamc@28 287
adamc@28 288 reflexivity.
adamc@28 289
adamc@28 290 (** Our second subgoal is more work and also demonstrates our first inductive hypothesis.
adamc@28 291
adamc@28 292 [[
adamc@28 293 n : nat
adamc@28 294 IHn : plus n O = n
adamc@28 295 ============================
adamc@28 296 plus (S n) O = S n
adamc@208 297
adamc@28 298 ]]
adamc@28 299
adam@315 300 We can start out by using computation to simplify the goal as far as we can.%\index{tactics!simpl}% *)
adamc@28 301
adamc@28 302 simpl.
adamc@28 303
adam@400 304 (** Now the conclusion is [S (plus n O) = S n]. Using our inductive hypothesis: *)
adamc@28 305
adamc@28 306 rewrite IHn.
adamc@28 307
adamc@28 308 (** ...we get a trivial conclusion [S n = S n]. *)
adamc@28 309
adamc@28 310 reflexivity.
adamc@28 311
adam@315 312 (** Not much really went on in this proof, so the [crush] tactic from the [CpdtTactics] module can prove this theorem automatically. *)
adamc@28 313
adamc@28 314 Restart.
adam@315 315
adamc@28 316 induction n; crush.
adamc@28 317 Qed.
adamc@41 318 (* end thide *)
adamc@28 319
adamc@28 320 (** We can check out the induction principle at work here: *)
adamc@28 321
adamc@28 322 Check nat_ind.
adamc@208 323 (** %\vspace{-.15in}% [[
adamc@208 324 nat_ind : forall P : nat -> Prop,
adamc@208 325 P O -> (forall n : nat, P n -> P (S n)) -> forall n : nat, P n
adamc@208 326 ]]
adamc@28 327
adam@439 328 %\smallskip{}%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@28 329
adam@315 330 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@28 331
adamc@28 332 Theorem S_inj : forall n m : nat, S n = S m -> n = m.
adamc@41 333 (* begin thide *)
adamc@28 334 injection 1; trivial.
adamc@28 335 Qed.
adamc@41 336 (* end thide *)
adamc@28 337
adam@400 338 (** 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@28 339
adam@417 340 There is also a very useful tactic called %\index{tactics!congruence}%[congruence] that can prove this theorem immediately. The [congruence] tactic 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@29 341
adamc@29 342 %\medskip%
adamc@29 343
adamc@29 344 We can define a type of lists of natural numbers. *)
adamc@29 345
adamc@29 346 Inductive nat_list : Set :=
adamc@29 347 | NNil : nat_list
adamc@29 348 | NCons : nat -> nat_list -> nat_list.
adamc@29 349
adamc@29 350 (** Recursive definitions are straightforward extensions of what we have seen before. *)
adamc@29 351
adamc@29 352 Fixpoint nlength (ls : nat_list) : nat :=
adamc@29 353 match ls with
adamc@29 354 | NNil => O
adamc@29 355 | NCons _ ls' => S (nlength ls')
adamc@29 356 end.
adamc@29 357
adamc@208 358 Fixpoint napp (ls1 ls2 : nat_list) : nat_list :=
adamc@29 359 match ls1 with
adamc@29 360 | NNil => ls2
adamc@29 361 | NCons n ls1' => NCons n (napp ls1' ls2)
adamc@29 362 end.
adamc@29 363
adamc@29 364 (** Inductive theorem proving can again be automated quite effectively. *)
adamc@29 365
adamc@29 366 Theorem nlength_napp : forall ls1 ls2 : nat_list, nlength (napp ls1 ls2)
adamc@29 367 = plus (nlength ls1) (nlength ls2).
adamc@41 368 (* begin thide *)
adamc@29 369 induction ls1; crush.
adamc@29 370 Qed.
adamc@41 371 (* end thide *)
adamc@29 372
adamc@29 373 Check nat_list_ind.
adamc@208 374 (** %\vspace{-.15in}% [[
adamc@208 375 nat_list_ind
adamc@29 376 : forall P : nat_list -> Prop,
adamc@29 377 P NNil ->
adamc@29 378 (forall (n : nat) (n0 : nat_list), P n0 -> P (NCons n n0)) ->
adamc@29 379 forall n : nat_list, P n
adamc@29 380 ]]
adamc@29 381
adamc@29 382 %\medskip%
adamc@29 383
adam@420 384 In general, we can implement any "tree" types as inductive types. For example, here are binary trees of naturals. *)
adamc@29 385
adamc@29 386 Inductive nat_btree : Set :=
adamc@29 387 | NLeaf : nat_btree
adamc@29 388 | NNode : nat_btree -> nat -> nat_btree -> nat_btree.
adamc@29 389
adamc@29 390 Fixpoint nsize (tr : nat_btree) : nat :=
adamc@29 391 match tr with
adamc@35 392 | NLeaf => S O
adamc@29 393 | NNode tr1 _ tr2 => plus (nsize tr1) (nsize tr2)
adamc@29 394 end.
adamc@29 395
adamc@208 396 Fixpoint nsplice (tr1 tr2 : nat_btree) : nat_btree :=
adamc@29 397 match tr1 with
adamc@35 398 | NLeaf => NNode tr2 O NLeaf
adamc@29 399 | NNode tr1' n tr2' => NNode (nsplice tr1' tr2) n tr2'
adamc@29 400 end.
adamc@29 401
adamc@29 402 Theorem plus_assoc : forall n1 n2 n3 : nat, plus (plus n1 n2) n3 = plus n1 (plus n2 n3).
adamc@41 403 (* begin thide *)
adamc@29 404 induction n1; crush.
adamc@29 405 Qed.
adamc@41 406 (* end thide *)
adamc@29 407
adamc@29 408 Theorem nsize_nsplice : forall tr1 tr2 : nat_btree, nsize (nsplice tr1 tr2)
adamc@29 409 = plus (nsize tr2) (nsize tr1).
adamc@41 410 (* begin thide *)
adam@375 411 Hint Rewrite n_plus_O plus_assoc.
adamc@29 412
adamc@29 413 induction tr1; crush.
adamc@29 414 Qed.
adamc@41 415 (* end thide *)
adamc@29 416
adam@315 417 (** 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@315 418
adamc@29 419 Check nat_btree_ind.
adamc@208 420 (** %\vspace{-.15in}% [[
adamc@208 421 nat_btree_ind
adamc@29 422 : forall P : nat_btree -> Prop,
adamc@29 423 P NLeaf ->
adamc@29 424 (forall n : nat_btree,
adamc@29 425 P n -> forall (n0 : nat) (n1 : nat_btree), P n1 -> P (NNode n n0 n1)) ->
adamc@29 426 forall n : nat_btree, P n
adam@302 427 ]]
adam@315 428
adam@315 429 We have the usual two cases, one for each constructor of [nat_btree]. *)
adamc@30 430
adamc@30 431
adamc@30 432 (** * Parameterized Types *)
adamc@30 433
adam@316 434 (** 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@30 435
adamc@30 436 Inductive list (T : Set) : Set :=
adamc@30 437 | Nil : list T
adamc@30 438 | Cons : T -> list T -> list T.
adamc@30 439
adamc@30 440 Fixpoint length T (ls : list T) : nat :=
adamc@30 441 match ls with
adamc@30 442 | Nil => O
adamc@30 443 | Cons _ ls' => S (length ls')
adamc@30 444 end.
adamc@30 445
adamc@208 446 Fixpoint app T (ls1 ls2 : list T) : list T :=
adamc@30 447 match ls1 with
adamc@30 448 | Nil => ls2
adamc@30 449 | Cons x ls1' => Cons x (app ls1' ls2)
adamc@30 450 end.
adamc@30 451
adamc@30 452 Theorem length_app : forall T (ls1 ls2 : list T), length (app ls1 ls2)
adamc@30 453 = plus (length ls1) (length ls2).
adamc@41 454 (* begin thide *)
adamc@30 455 induction ls1; crush.
adamc@30 456 Qed.
adamc@41 457 (* end thide *)
adamc@30 458
adam@420 459 (** 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@30 460
adamc@30 461 (* begin hide *)
adamc@30 462 Reset list.
adamc@30 463 (* end hide *)
adamc@30 464
adamc@30 465 Section list.
adamc@30 466 Variable T : Set.
adamc@30 467
adamc@30 468 Inductive list : Set :=
adamc@30 469 | Nil : list
adamc@30 470 | Cons : T -> list -> list.
adamc@30 471
adamc@30 472 Fixpoint length (ls : list) : nat :=
adamc@30 473 match ls with
adamc@30 474 | Nil => O
adamc@30 475 | Cons _ ls' => S (length ls')
adamc@30 476 end.
adamc@30 477
adamc@208 478 Fixpoint app (ls1 ls2 : list) : list :=
adamc@30 479 match ls1 with
adamc@30 480 | Nil => ls2
adamc@30 481 | Cons x ls1' => Cons x (app ls1' ls2)
adamc@30 482 end.
adamc@30 483
adamc@30 484 Theorem length_app : forall ls1 ls2 : list, length (app ls1 ls2)
adamc@30 485 = plus (length ls1) (length ls2).
adamc@41 486 (* begin thide *)
adamc@30 487 induction ls1; crush.
adamc@30 488 Qed.
adamc@41 489 (* end thide *)
adamc@30 490 End list.
adamc@30 491
adamc@35 492 (* begin hide *)
adamc@35 493 Implicit Arguments Nil [T].
adamc@35 494 (* end hide *)
adamc@35 495
adamc@210 496 (** 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@30 497
adamc@202 498 Print list.
adamc@208 499 (** %\vspace{-.15in}% [[
adamc@208 500 Inductive list (T : Set) : Set :=
adam@316 501 Nil : list T | Cons : T -> list T -> list T
adamc@202 502 ]]
adamc@30 503
adam@439 504 %\smallskip{}%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@30 505
adamc@30 506 Check length.
adamc@208 507 (** %\vspace{-.15in}% [[
adamc@208 508 length
adamc@30 509 : forall T : Set, list T -> nat
adamc@30 510 ]]
adamc@30 511
adam@439 512 %\smallskip{}%The parameter [T] is treated as a new argument to the induction principle, too. *)
adamc@30 513
adamc@30 514 Check list_ind.
adamc@208 515 (** %\vspace{-.15in}% [[
adamc@208 516 list_ind
adamc@30 517 : forall (T : Set) (P : list T -> Prop),
adamc@30 518 P (Nil T) ->
adamc@30 519 (forall (t : T) (l : list T), P l -> P (Cons t l)) ->
adamc@30 520 forall l : list T, P l
adamc@30 521 ]]
adamc@30 522
adam@439 523 %\smallskip{}%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@31 524
adamc@31 525
adamc@31 526 (** * Mutually Inductive Types *)
adamc@31 527
adamc@31 528 (** We can define inductive types that refer to each other: *)
adamc@31 529
adamc@31 530 Inductive even_list : Set :=
adamc@31 531 | ENil : even_list
adamc@31 532 | ECons : nat -> odd_list -> even_list
adamc@31 533
adamc@31 534 with odd_list : Set :=
adamc@31 535 | OCons : nat -> even_list -> odd_list.
adamc@31 536
adamc@31 537 Fixpoint elength (el : even_list) : nat :=
adamc@31 538 match el with
adamc@31 539 | ENil => O
adamc@31 540 | ECons _ ol => S (olength ol)
adamc@31 541 end
adamc@31 542
adamc@31 543 with olength (ol : odd_list) : nat :=
adamc@31 544 match ol with
adamc@31 545 | OCons _ el => S (elength el)
adamc@31 546 end.
adamc@31 547
adamc@208 548 Fixpoint eapp (el1 el2 : even_list) : even_list :=
adamc@31 549 match el1 with
adamc@31 550 | ENil => el2
adamc@31 551 | ECons n ol => ECons n (oapp ol el2)
adamc@31 552 end
adamc@31 553
adamc@208 554 with oapp (ol : odd_list) (el : even_list) : odd_list :=
adamc@31 555 match ol with
adamc@31 556 | OCons n el' => OCons n (eapp el' el)
adamc@31 557 end.
adamc@31 558
adamc@31 559 (** Everything is going roughly the same as in past examples, until we try to prove a theorem similar to those that came before. *)
adamc@31 560
adamc@31 561 Theorem elength_eapp : forall el1 el2 : even_list,
adamc@31 562 elength (eapp el1 el2) = plus (elength el1) (elength el2).
adamc@41 563 (* begin thide *)
adamc@31 564 induction el1; crush.
adamc@31 565
adamc@31 566 (** One goal remains: [[
adamc@31 567
adamc@31 568 n : nat
adamc@31 569 o : odd_list
adamc@31 570 el2 : even_list
adamc@31 571 ============================
adamc@31 572 S (olength (oapp o el2)) = S (plus (olength o) (elength el2))
adamc@31 573 ]]
adamc@31 574
adamc@31 575 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@31 576
adamc@31 577 Abort.
adamc@31 578 Check even_list_ind.
adamc@208 579 (** %\vspace{-.15in}% [[
adamc@208 580 even_list_ind
adamc@31 581 : forall P : even_list -> Prop,
adamc@31 582 P ENil ->
adamc@31 583 (forall (n : nat) (o : odd_list), P (ECons n o)) ->
adamc@31 584 forall e : even_list, P e
adamc@31 585 ]]
adamc@31 586
adam@439 587 %\smallskip{}%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@31 588
adamc@31 589 Scheme even_list_mut := Induction for even_list Sort Prop
adamc@31 590 with odd_list_mut := Induction for odd_list Sort Prop.
adamc@31 591
adam@316 592 (** 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@316 593
adamc@31 594 Check even_list_mut.
adamc@208 595 (** %\vspace{-.15in}% [[
adamc@208 596 even_list_mut
adamc@31 597 : forall (P : even_list -> Prop) (P0 : odd_list -> Prop),
adamc@31 598 P ENil ->
adamc@31 599 (forall (n : nat) (o : odd_list), P0 o -> P (ECons n o)) ->
adamc@31 600 (forall (n : nat) (e : even_list), P e -> P0 (OCons n e)) ->
adamc@31 601 forall e : even_list, P e
adamc@208 602
adamc@31 603 ]]
adamc@31 604
adam@316 605 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@31 606
adamc@31 607 Theorem n_plus_O' : forall n : nat, plus n O = n.
adamc@31 608 apply (nat_ind (fun n => plus n O = n)); crush.
adamc@31 609 Qed.
adamc@31 610
adamc@31 611 (** 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@31 612
adamc@31 613 This technique generalizes to our mutual example: *)
adamc@31 614
adamc@31 615 Theorem elength_eapp : forall el1 el2 : even_list,
adamc@31 616 elength (eapp el1 el2) = plus (elength el1) (elength el2).
adamc@41 617
adamc@31 618 apply (even_list_mut
adamc@31 619 (fun el1 : even_list => forall el2 : even_list,
adamc@31 620 elength (eapp el1 el2) = plus (elength el1) (elength el2))
adamc@31 621 (fun ol : odd_list => forall el : even_list,
adamc@31 622 olength (oapp ol el) = plus (olength ol) (elength el))); crush.
adamc@31 623 Qed.
adamc@41 624 (* end thide *)
adamc@31 625
adamc@31 626 (** 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@33 627
adamc@33 628
adamc@33 629 (** * Reflexive Types *)
adamc@33 630
adam@398 631 (** 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@316 632
adam@316 633 Inductive pformula : Set :=
adam@316 634 | Truth : pformula
adam@316 635 | Falsehood : pformula
adam@316 636 | Conjunction : pformula -> pformula -> pformula.
adam@316 637
adam@398 638 (** 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@316 639
adam@316 640 Fixpoint pformulaDenote (f : pformula) : Prop :=
adam@316 641 match f with
adam@316 642 | Truth => True
adam@316 643 | Falsehood => False
adam@316 644 | Conjunction f1 f2 => pformulaDenote f1 /\ pformulaDenote f2
adam@316 645 end.
adam@316 646
adam@392 647 (** 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@33 648
adamc@33 649 Inductive formula : Set :=
adamc@33 650 | Eq : nat -> nat -> formula
adamc@33 651 | And : formula -> formula -> formula
adamc@33 652 | Forall : (nat -> formula) -> formula.
adamc@33 653
adam@420 654 (** 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@33 655
adamc@33 656 Example forall_refl : formula := Forall (fun x => Eq x x).
adamc@33 657
adamc@33 658 (** We can write recursive functions over reflexive types quite naturally. Here is one translating our formulas into native Coq propositions. *)
adamc@33 659
adamc@33 660 Fixpoint formulaDenote (f : formula) : Prop :=
adamc@33 661 match f with
adamc@33 662 | Eq n1 n2 => n1 = n2
adamc@33 663 | And f1 f2 => formulaDenote f1 /\ formulaDenote f2
adamc@33 664 | Forall f' => forall n : nat, formulaDenote (f' n)
adamc@33 665 end.
adamc@33 666
adamc@33 667 (** We can also encode a trivial formula transformation that swaps the order of equality and conjunction operands. *)
adamc@33 668
adamc@33 669 Fixpoint swapper (f : formula) : formula :=
adamc@33 670 match f with
adamc@33 671 | Eq n1 n2 => Eq n2 n1
adamc@33 672 | And f1 f2 => And (swapper f2) (swapper f1)
adamc@33 673 | Forall f' => Forall (fun n => swapper (f' n))
adamc@33 674 end.
adamc@33 675
adamc@33 676 (** It is helpful to prove that this transformation does not make true formulas false. *)
adamc@33 677
adamc@33 678 Theorem swapper_preserves_truth : forall f, formulaDenote f -> formulaDenote (swapper f).
adamc@41 679 (* begin thide *)
adamc@33 680 induction f; crush.
adamc@33 681 Qed.
adamc@41 682 (* end thide *)
adamc@33 683
adamc@33 684 (** We can take a look at the induction principle behind this proof. *)
adamc@33 685
adamc@33 686 Check formula_ind.
adamc@208 687 (** %\vspace{-.15in}% [[
adamc@208 688 formula_ind
adamc@33 689 : forall P : formula -> Prop,
adamc@33 690 (forall n n0 : nat, P (Eq n n0)) ->
adamc@33 691 (forall f0 : formula,
adamc@33 692 P f0 -> forall f1 : formula, P f1 -> P (And f0 f1)) ->
adamc@33 693 (forall f1 : nat -> formula,
adamc@33 694 (forall n : nat, P (f1 n)) -> P (Forall f1)) ->
adamc@33 695 forall f2 : formula, P f2
adamc@208 696 ]]
adamc@33 697
adam@439 698 %\smallskip{}%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@33 699
adamc@33 700 %\medskip%
adamc@33 701
adam@316 702 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@33 703
adam@400 704 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@400 705 (* begin hide *)
adam@437 706 (* begin thide *)
adam@400 707 Inductive term : Set := App | Abs.
adam@400 708 Reset term.
adam@420 709 Definition uhoh := O.
adam@437 710 (* end thide *)
adam@400 711 (* end hide *)
adamc@33 712 (** [[
adamc@33 713 Inductive term : Set :=
adamc@33 714 | App : term -> term -> term
adamc@33 715 | Abs : (term -> term) -> term.
adamc@33 716 ]]
adamc@33 717
adam@316 718 <<
adam@316 719 Error: Non strictly positive occurrence of "term" in "(term -> term) -> term"
adam@316 720 >>
adam@316 721
adam@400 722 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@33 723
adamc@33 724 Why must Coq enforce this restriction? Imagine that our last definition had been accepted, allowing us to write this function:
adamc@33 725
adam@439 726 %\vspace{-.15in}%[[
adamc@33 727 Definition uhoh (t : term) : term :=
adamc@33 728 match t with
adamc@33 729 | Abs f => f t
adamc@33 730 | _ => t
adamc@33 731 end.
adamc@205 732 ]]
adamc@205 733
adamc@33 734 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@33 735
adam@316 736 %\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@33 737
adam@439 738 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 final chapter, on programming language syntax and semantics. *)
adamc@34 739
adamc@34 740
adam@317 741 (** * An Interlude on Induction Principles *)
adamc@34 742
adam@317 743 (** 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@34 744
adamc@34 745 Print unit_ind.
adam@437 746 (** %\vspace{-.15in}%[[
adamc@208 747 unit_ind =
adamc@208 748 fun P : unit -> Prop => unit_rect P
adamc@34 749 : forall P : unit -> Prop, P tt -> forall u : unit, P u
adamc@34 750 ]]
adamc@34 751
adam@439 752 %\smallskip{}%We see that this induction principle is defined in terms of a more general principle, [unit_rect]. The <<rec>> stands for "recursion principle," and the <<t>> at the end stands for [Type]. *)
adamc@34 753
adamc@34 754 Check unit_rect.
adamc@208 755 (** %\vspace{-.15in}% [[
adamc@208 756 unit_rect
adamc@34 757 : forall P : unit -> Type, P tt -> forall u : unit, P u
adamc@34 758 ]]
adamc@34 759
adam@439 760 %\smallskip{}%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@34 761
adamc@34 762 Print unit_rec.
adam@437 763 (** %\vspace{-.15in}%[[
adamc@208 764 unit_rec =
adamc@208 765 fun P : unit -> Set => unit_rect P
adamc@34 766 : forall P : unit -> Set, P tt -> forall u : unit, P u
adamc@34 767 ]]
adamc@34 768
adam@439 769 %\smallskip{}%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@34 770
adamc@34 771 Definition always_O (u : unit) : nat :=
adamc@34 772 match u with
adamc@34 773 | tt => O
adamc@34 774 end.
adamc@34 775
adamc@34 776 Definition always_O' (u : unit) : nat :=
adamc@34 777 unit_rec (fun _ : unit => nat) O u.
adamc@34 778
adamc@34 779 (** 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@34 780
adamc@34 781 Print unit_rect.
adam@437 782 (** %\vspace{-.15in}%[[
adamc@208 783 unit_rect =
adamc@208 784 fun (P : unit -> Type) (f : P tt) (u : unit) =>
adamc@208 785 match u as u0 return (P u0) with
adamc@208 786 | tt => f
adamc@208 787 end
adamc@34 788 : forall P : unit -> Type, P tt -> forall u : unit, P u
adamc@34 789 ]]
adamc@34 790
adam@439 791 %\smallskip{}%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@317 792
adam@317 793 %\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@34 794
adamc@34 795 To prove that [unit_rect] is nothing special, we can reimplement it manually. *)
adamc@34 796
adamc@34 797 Definition unit_rect' (P : unit -> Type) (f : P tt) (u : unit) :=
adamc@208 798 match u with
adamc@34 799 | tt => f
adamc@34 800 end.
adamc@34 801
adam@420 802 (* begin hide *)
adam@437 803 (* begin thide *)
adam@420 804 Definition foo := nat_rect.
adam@437 805 (* end thide *)
adam@420 806 (* end hide *)
adam@420 807
adam@317 808 (** We rely on Coq's heuristics for inferring [match] annotations, which are not consulted in the pretty-printing of terms.
adamc@34 809
adam@400 810 We can check the implementation [nat_rect] as well: *)
adamc@34 811
adamc@34 812 Print nat_rect.
adam@439 813 (** %\vspace{-.15in}% [[
adam@400 814 nat_rect =
adamc@208 815 fun (P : nat -> Type) (f : P O) (f0 : forall n : nat, P n -> P (S n)) =>
adamc@208 816 fix F (n : nat) : P n :=
adamc@208 817 match n as n0 return (P n0) with
adamc@208 818 | O => f
adamc@208 819 | S n0 => f0 n0 (F n0)
adamc@208 820 end
adamc@208 821 : forall P : nat -> Type,
adamc@208 822 P O -> (forall n : nat, P n -> P (S n)) -> forall n : nat, P n
adamc@208 823 ]]
adamc@34 824
adam@439 825 %\smallskip{}%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@34 826
adam@317 827 Section nat_ind'.
adamc@208 828 (** First, we have the property of natural numbers that we aim to prove. *)
adamc@34 829
adam@317 830 Variable P : nat -> Prop.
adamc@34 831
adam@317 832 (** 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@34 833
adam@317 834 Hypothesis O_case : P O.
adamc@34 835
adamc@208 836 (** Next is a proof of the [S] case, which may assume an inductive hypothesis. *)
adamc@34 837
adam@317 838 Hypothesis S_case : forall n : nat, P n -> P (S n).
adamc@34 839
adamc@208 840 (** Finally, we define a recursive function to tie the pieces together. *)
adamc@34 841
adam@317 842 Fixpoint nat_ind' (n : nat) : P n :=
adam@317 843 match n with
adam@317 844 | O => O_case
adam@317 845 | S n' => S_case (nat_ind' n')
adam@317 846 end.
adam@317 847 End nat_ind'.
adamc@34 848
adam@400 849 (** 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@34 850
adam@317 851 %\medskip%
adamc@34 852
adam@317 853 We can also examine the definition of [even_list_mut], which we generated with [Scheme] for a mutually recursive type. *)
adamc@34 854
adam@317 855 Print even_list_mut.
adam@439 856 (** %\vspace{-.15in}%[[
adam@317 857 even_list_mut =
adam@317 858 fun (P : even_list -> Prop) (P0 : odd_list -> Prop)
adam@317 859 (f : P ENil) (f0 : forall (n : nat) (o : odd_list), P0 o -> P (ECons n o))
adam@317 860 (f1 : forall (n : nat) (e : even_list), P e -> P0 (OCons n e)) =>
adam@317 861 fix F (e : even_list) : P e :=
adam@317 862 match e as e0 return (P e0) with
adam@317 863 | ENil => f
adam@317 864 | ECons n o => f0 n o (F0 o)
adam@317 865 end
adam@317 866 with F0 (o : odd_list) : P0 o :=
adam@317 867 match o as o0 return (P0 o0) with
adam@317 868 | OCons n e => f1 n e (F e)
adam@317 869 end
adam@317 870 for F
adam@317 871 : forall (P : even_list -> Prop) (P0 : odd_list -> Prop),
adam@317 872 P ENil ->
adam@317 873 (forall (n : nat) (o : odd_list), P0 o -> P (ECons n o)) ->
adam@317 874 (forall (n : nat) (e : even_list), P e -> P0 (OCons n e)) ->
adam@317 875 forall e : even_list, P e
adam@317 876 ]]
adamc@34 877
adam@439 878 %\smallskip{}%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 <<and>> 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@208 879
adam@317 880 Section even_list_mut'.
adam@317 881 (** First, we need the properties that we are proving. *)
adamc@208 882
adam@317 883 Variable Peven : even_list -> Prop.
adam@317 884 Variable Podd : odd_list -> Prop.
adamc@208 885
adam@317 886 (** Next, we need proofs of the three cases. *)
adamc@208 887
adam@317 888 Hypothesis ENil_case : Peven ENil.
adam@317 889 Hypothesis ECons_case : forall (n : nat) (o : odd_list), Podd o -> Peven (ECons n o).
adam@317 890 Hypothesis OCons_case : forall (n : nat) (e : even_list), Peven e -> Podd (OCons n e).
adamc@208 891
adam@317 892 (** Finally, we define the recursive functions. *)
adamc@208 893
adam@317 894 Fixpoint even_list_mut' (e : even_list) : Peven e :=
adam@317 895 match e with
adam@317 896 | ENil => ENil_case
adam@317 897 | ECons n o => ECons_case n (odd_list_mut' o)
adam@317 898 end
adam@317 899 with odd_list_mut' (o : odd_list) : Podd o :=
adam@317 900 match o with
adam@317 901 | OCons n e => OCons_case n (even_list_mut' e)
adam@317 902 end.
adamc@34 903 End even_list_mut'.
adamc@34 904
adamc@34 905 (** 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@34 906
adamc@34 907 Section formula_ind'.
adamc@34 908 Variable P : formula -> Prop.
adamc@38 909 Hypothesis Eq_case : forall n1 n2 : nat, P (Eq n1 n2).
adamc@38 910 Hypothesis And_case : forall f1 f2 : formula,
adamc@34 911 P f1 -> P f2 -> P (And f1 f2).
adamc@38 912 Hypothesis Forall_case : forall f : nat -> formula,
adamc@34 913 (forall n : nat, P (f n)) -> P (Forall f).
adamc@34 914
adamc@34 915 Fixpoint formula_ind' (f : formula) : P f :=
adamc@208 916 match f with
adamc@34 917 | Eq n1 n2 => Eq_case n1 n2
adamc@34 918 | And f1 f2 => And_case (formula_ind' f1) (formula_ind' f2)
adamc@34 919 | Forall f' => Forall_case f' (fun n => formula_ind' (f' n))
adamc@34 920 end.
adamc@34 921 End formula_ind'.
adamc@34 922
adam@317 923 (** It is apparent that induction principle implementations involve some tedium but not terribly much creativity. *)
adam@317 924
adamc@35 925
adamc@35 926 (** * Nested Inductive Types *)
adamc@35 927
adamc@35 928 (** 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@35 929
adamc@35 930 Inductive nat_tree : Set :=
adamc@35 931 | NLeaf' : nat_tree
adamc@35 932 | NNode' : nat -> list nat_tree -> nat_tree.
adamc@35 933
adam@420 934 (** 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@35 935
adam@317 936 Like we encountered for mutual inductive types, we find that the automatically generated induction principle for [nat_tree] is too weak. *)
adamc@35 937
adamc@35 938 Check nat_tree_ind.
adamc@208 939 (** %\vspace{-.15in}% [[
adamc@208 940 nat_tree_ind
adamc@35 941 : forall P : nat_tree -> Prop,
adamc@35 942 P NLeaf' ->
adamc@35 943 (forall (n : nat) (l : list nat_tree), P (NNode' n l)) ->
adamc@35 944 forall n : nat_tree, P n
adamc@35 945 ]]
adamc@35 946
adam@439 947 %\smallskip{}%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@35 948
adamc@35 949 First, we will need an auxiliary definition, characterizing what it means for a property to hold of every element of a list. *)
adamc@35 950
adamc@35 951 Section All.
adamc@35 952 Variable T : Set.
adamc@35 953 Variable P : T -> Prop.
adamc@35 954
adamc@35 955 Fixpoint All (ls : list T) : Prop :=
adamc@35 956 match ls with
adamc@35 957 | Nil => True
adamc@35 958 | Cons h t => P h /\ All t
adamc@35 959 end.
adamc@35 960 End All.
adamc@35 961
adam@317 962 (** It will be useful to review the definitions of [True] and [/\], since we will want to write manual proofs of them below. *)
adamc@35 963
adamc@35 964 Print True.
adam@439 965 (** %\vspace{-.15in}%[[
adamc@208 966 Inductive True : Prop := I : True
adamc@208 967 ]]
adamc@35 968
adam@439 969 %\smallskip{}%That is, [True] is a proposition with exactly one proof, [I], which we may always supply trivially.
adamc@35 970
adam@400 971 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@35 972
adamc@35 973 Locate "/\".
adam@439 974 (** %\vspace{-.15in}%[[
adam@317 975 "A /\ B" := and A B : type_scope (default interpretation)
adam@302 976 ]]
adam@302 977 *)
adamc@35 978
adamc@35 979 Print and.
adam@439 980 (** %\vspace{-.15in}%[[
adamc@208 981 Inductive and (A : Prop) (B : Prop) : Prop := conj : A -> B -> A /\ B
adam@317 982 ]]
adam@317 983 %\vspace{-.1in}%
adam@317 984 <<
adamc@208 985 For conj: Arguments A, B are implicit
adam@317 986 >>
adamc@35 987
adam@400 988 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@35 989
adamc@35 990 %\medskip%
adamc@35 991
adamc@35 992 Now we create a section for our induction principle, following the same basic plan as in the last section of this chapter. *)
adamc@35 993
adamc@35 994 Section nat_tree_ind'.
adamc@35 995 Variable P : nat_tree -> Prop.
adamc@35 996
adamc@38 997 Hypothesis NLeaf'_case : P NLeaf'.
adamc@38 998 Hypothesis NNode'_case : forall (n : nat) (ls : list nat_tree),
adamc@35 999 All P ls -> P (NNode' n ls).
adamc@35 1000
adam@420 1001 (* begin hide *)
adam@437 1002 (* begin thide *)
adam@420 1003 Definition list_nat_tree_ind := O.
adam@437 1004 (* end thide *)
adam@420 1005 (* end hide *)
adam@420 1006
adamc@35 1007 (** 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 1008
adam@439 1009 %\vspace{-.15in}%[[
adamc@35 1010 Fixpoint nat_tree_ind' (tr : nat_tree) : P tr :=
adamc@208 1011 match tr with
adamc@35 1012 | NLeaf' => NLeaf'_case
adamc@35 1013 | NNode' n ls => NNode'_case n ls (list_nat_tree_ind ls)
adamc@35 1014 end
adamc@35 1015
adamc@35 1016 with list_nat_tree_ind (ls : list nat_tree) : All P ls :=
adamc@208 1017 match ls with
adamc@35 1018 | Nil => I
adamc@35 1019 | Cons tr rest => conj (nat_tree_ind' tr) (list_nat_tree_ind rest)
adamc@35 1020 end.
adamc@205 1021 ]]
adamc@205 1022
adam@439 1023 %\smallskip{}%Coq rejects this definition, saying
adam@317 1024 <<
adam@317 1025 Recursive call to nat_tree_ind' has principal argument equal to "tr"
adam@317 1026 instead of rest.
adam@317 1027 >>
adam@317 1028
adam@420 1029 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 1030
adamc@35 1031 Fixpoint nat_tree_ind' (tr : nat_tree) : P tr :=
adamc@208 1032 match tr with
adamc@35 1033 | NLeaf' => NLeaf'_case
adamc@35 1034 | NNode' n ls => NNode'_case n ls
adamc@35 1035 ((fix list_nat_tree_ind (ls : list nat_tree) : All P ls :=
adamc@208 1036 match ls with
adamc@35 1037 | Nil => I
adamc@35 1038 | Cons tr rest => conj (nat_tree_ind' tr) (list_nat_tree_ind rest)
adamc@35 1039 end) ls)
adamc@35 1040 end.
adamc@35 1041
adam@398 1042 (** 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 1043
adamc@35 1044 End nat_tree_ind'.
adamc@35 1045
adamc@35 1046 (** 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 1047
adamc@35 1048 Section map.
adamc@35 1049 Variables T T' : Set.
adam@317 1050 Variable F : T -> T'.
adamc@35 1051
adamc@35 1052 Fixpoint map (ls : list T) : list T' :=
adamc@35 1053 match ls with
adamc@35 1054 | Nil => Nil
adam@317 1055 | Cons h t => Cons (F h) (map t)
adamc@35 1056 end.
adamc@35 1057 End map.
adamc@35 1058
adamc@35 1059 Fixpoint sum (ls : list nat) : nat :=
adamc@35 1060 match ls with
adamc@35 1061 | Nil => O
adamc@35 1062 | Cons h t => plus h (sum t)
adamc@35 1063 end.
adamc@35 1064
adamc@35 1065 (** Now we can define a size function over our trees. *)
adamc@35 1066
adamc@35 1067 Fixpoint ntsize (tr : nat_tree) : nat :=
adamc@35 1068 match tr with
adamc@35 1069 | NLeaf' => S O
adamc@35 1070 | NNode' _ trs => S (sum (map ntsize trs))
adamc@35 1071 end.
adamc@35 1072
adamc@35 1073 (** 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 1074
adamc@208 1075 Fixpoint ntsplice (tr1 tr2 : nat_tree) : nat_tree :=
adamc@35 1076 match tr1 with
adamc@35 1077 | NLeaf' => NNode' O (Cons tr2 Nil)
adamc@35 1078 | NNode' n Nil => NNode' n (Cons tr2 Nil)
adamc@35 1079 | NNode' n (Cons tr trs) => NNode' n (Cons (ntsplice tr tr2) trs)
adamc@35 1080 end.
adamc@35 1081
adamc@35 1082 (** 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 1083
adamc@41 1084 (* begin thide *)
adamc@35 1085 Lemma plus_S : forall n1 n2 : nat,
adamc@35 1086 plus n1 (S n2) = S (plus n1 n2).
adamc@35 1087 induction n1; crush.
adamc@35 1088 Qed.
adamc@41 1089 (* end thide *)
adamc@35 1090
adamc@35 1091 (** Now we begin the proof of the theorem, adding the lemma [plus_S] as a hint. *)
adamc@35 1092
adamc@35 1093 Theorem ntsize_ntsplice : forall tr1 tr2 : nat_tree, ntsize (ntsplice tr1 tr2)
adamc@35 1094 = plus (ntsize tr2) (ntsize tr1).
adamc@41 1095 (* begin thide *)
adam@375 1096 Hint Rewrite plus_S.
adamc@35 1097
adam@317 1098 (** 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 1099
adamc@35 1100 induction tr1 using nat_tree_ind'; crush.
adamc@35 1101
adamc@35 1102 (** One subgoal remains: [[
adamc@35 1103 n : nat
adamc@35 1104 ls : list nat_tree
adamc@35 1105 H : All
adamc@35 1106 (fun tr1 : nat_tree =>
adamc@35 1107 forall tr2 : nat_tree,
adamc@35 1108 ntsize (ntsplice tr1 tr2) = plus (ntsize tr2) (ntsize tr1)) ls
adamc@35 1109 tr2 : nat_tree
adamc@35 1110 ============================
adamc@35 1111 ntsize
adamc@35 1112 match ls with
adamc@35 1113 | Nil => NNode' n (Cons tr2 Nil)
adamc@35 1114 | Cons tr trs => NNode' n (Cons (ntsplice tr tr2) trs)
adamc@35 1115 end = S (plus (ntsize tr2) (sum (map ntsize ls)))
adamc@208 1116
adamc@35 1117 ]]
adamc@35 1118
adamc@35 1119 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 1120
adamc@35 1121 destruct ls; crush.
adamc@35 1122
adam@317 1123 (** We can go further in automating the proof by exploiting the hint mechanism.%\index{Vernacular commands!Hint Extern}% *)
adamc@35 1124
adamc@35 1125 Restart.
adam@317 1126
adamc@35 1127 Hint Extern 1 (ntsize (match ?LS with Nil => _ | Cons _ _ => _ end) = _) =>
adamc@35 1128 destruct LS; crush.
adamc@35 1129 induction tr1 using nat_tree_ind'; crush.
adamc@35 1130 Qed.
adamc@41 1131 (* end thide *)
adamc@35 1132
adamc@35 1133 (** 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 1134
adam@317 1135 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 1136
adamc@36 1137
adamc@36 1138 (** * Manual Proofs About Constructors *)
adamc@36 1139
adam@317 1140 (** 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 1141
adamc@36 1142 Theorem true_neq_false : true <> false.
adamc@208 1143
adamc@41 1144 (* begin thide *)
adam@420 1145 (** 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 1146
adamc@36 1147 red.
adam@439 1148 (** %\vspace{-.15in}%[[
adamc@36 1149 ============================
adamc@36 1150 true = false -> False
adamc@36 1151 ]]
adamc@36 1152
adam@439 1153 %\smallskip{}%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 1154
adamc@36 1155 intro H.
adam@439 1156 (** %\vspace{-.15in}%[[
adamc@36 1157 H : true = false
adamc@36 1158 ============================
adamc@36 1159 False
adamc@36 1160 ]]
adamc@36 1161
adam@439 1162 %\smallskip{}%This is the point in the proof where we apply some creativity. We define a function whose utility will become clear soon. *)
adamc@36 1163
adam@317 1164 Definition toProp (b : bool) := if b then True else False.
adamc@36 1165
adam@400 1166 (** 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 1167
adam@317 1168 change (toProp false).
adam@439 1169 (** %\vspace{-.15in}%[[
adamc@36 1170 H : true = false
adamc@36 1171 ============================
adam@317 1172 toProp false
adamc@36 1173 ]]
adamc@36 1174
adam@439 1175 %\smallskip{}%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 1176
adamc@36 1177 rewrite <- H.
adam@439 1178 (** %\vspace{-.15in}%[[
adamc@36 1179 H : true = false
adamc@36 1180 ============================
adam@317 1181 toProp true
adamc@36 1182 ]]
adamc@36 1183
adam@439 1184 %\smallskip{}%We are almost done. Just how close we are to done is revealed by computational simplification. *)
adamc@36 1185
adamc@36 1186 simpl.
adam@439 1187 (** %\vspace{-.15in}%[[
adamc@36 1188 H : true = false
adamc@36 1189 ============================
adamc@36 1190 True
adam@302 1191 ]]
adam@302 1192 *)
adamc@36 1193
adamc@36 1194 trivial.
adamc@36 1195 Qed.
adamc@41 1196 (* end thide *)
adamc@36 1197
adamc@36 1198 (** I have no trivial automated version of this proof to suggest, beyond using [discriminate] or [congruence] in the first place.
adamc@36 1199
adamc@36 1200 %\medskip%
adamc@36 1201
adamc@36 1202 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 1203
adamc@36 1204 Theorem S_inj' : forall n m : nat, S n = S m -> n = m.
adamc@41 1205 (* begin thide *)
adamc@36 1206 intros n m H.
adamc@36 1207 change (pred (S n) = pred (S m)).
adamc@36 1208 rewrite H.
adamc@36 1209 reflexivity.
adamc@36 1210 Qed.
adamc@41 1211 (* end thide *)
adamc@36 1212
adam@400 1213 (** 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 1214
adam@317 1215 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. *)