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author	Adam Chlipala <adam@chlipala.net>
date	Wed, 14 Sep 2011 14:04:08 -0400
parents	70e51e8cfce7
children	549d604c3d16

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adam@315	1 (* Copyright (c) 2008-2011, 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@315	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}\emph{%#<i>#Curry-Howard correspondence#</i>#%}~\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@315	68 (** Every one of these example programs whose type looks like a logical formula is a %\index{proof term}\emph{%#<i>#proof term#</i>#%}%. 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@317	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] %\emph{%#<i>#evaluates#</i>#%}% to either [true] or [false]. This means that we have an %\emph{%#<i>#algorithm#</i>#%}% 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@315	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
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@315	121 What exactly %\textit{%#<i>#is#</i>#%}% 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@315	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@315	153 (** [Empty_set_ind : forall (][P : Empty_set -> Prop) (e : Empty_set), P e] *)
adamc@26	154
adam@315	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
adam@315	210 (* begin hide *)
adamc@26	211 Restart.
adam@315	212 (* end hide *)
adam@315	213 (** %\noindent \coqdockw{Restart}%#<tt>Restart</tt>#. *)
adam@315	214
adamc@26	215 destruct b; reflexivity.
adamc@26	216 Qed.
adamc@41	217 (* end thide *)
adamc@27	218
adam@315	219 (** Another theorem about booleans illustrates another useful tactic.%\index{tactics!discriminate}% *)
adamc@27	220
adam@279	221 Theorem negb_ineq : forall b : bool, negb b <> b.
adamc@41	222 (* begin thide *)
adamc@27	223 destruct b; discriminate.
adamc@27	224 Qed.
adamc@41	225 (* end thide *)
adamc@27	226
adamc@27	227 (** [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	228
adamc@27	229 At this point, it is probably not hard to guess what the underlying induction principle for [bool] is. *)
adamc@27	230
adamc@27	231 Check bool_ind.
adamc@208	232 (** [bool_ind : forall P : bool -> Prop, P true -> P false -> forall b : bool, P b] *)
adamc@28	233
adam@315	234 (** That is, to prove that a property describes all [bool]s, prove that it describes both [true] and [false].
adam@315	235
adam@315	236 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 it 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	237
adamc@28	238
adamc@28	239 (** * Simple Recursive Types *)
adamc@28	240
adam@315	241 (** 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	242
adamc@28	243 Inductive nat : Set :=
adamc@28	244 \| O : nat
adamc@28	245 \| S : nat -> nat.
adamc@28	246
adam@315	247 (** [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	248
adam@316	249 Pattern matching works as we demonstrated in the last chapter:%\index{Gallina terms!pred}% *)
adamc@28	250
adamc@28	251 Definition isZero (n : nat) : bool :=
adamc@28	252 match n with
adamc@28	253 \| O => true
adamc@28	254 \| S _ => false
adamc@28	255 end.
adamc@28	256
adamc@28	257 Definition pred (n : nat) : nat :=
adamc@28	258 match n with
adamc@28	259 \| O => O
adamc@28	260 \| S n' => n'
adamc@28	261 end.
adamc@28	262
adamc@28	263 (** We can prove theorems by case analysis: *)
adamc@28	264
adamc@28	265 Theorem S_isZero : forall n : nat, isZero (pred (S (S n))) = false.
adamc@41	266 (* begin thide *)
adamc@28	267 destruct n; reflexivity.
adamc@28	268 Qed.
adamc@41	269 (* end thide *)
adamc@28	270
adam@316	271 (** 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	272
adamc@208	273 Fixpoint plus (n m : nat) : nat :=
adamc@28	274 match n with
adamc@28	275 \| O => m
adamc@28	276 \| S n' => S (plus n' m)
adamc@28	277 end.
adamc@28	278
adamc@208	279 (** Recall that [Fixpoint] is Coq's mechanism for recursive function definitions. Some theorems about [plus] can be proved without induction. *)
adamc@28	280
adamc@28	281 Theorem O_plus_n : forall n : nat, plus O n = n.
adamc@41	282 (* begin thide *)
adamc@28	283 intro; reflexivity.
adamc@28	284 Qed.
adamc@41	285 (* end thide *)
adamc@28	286
adamc@208	287 (** 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	288
adamc@28	289 Theorem n_plus_O : forall n : nat, plus n O = n.
adamc@41	290 (* begin thide *)
adamc@28	291 induction n.
adamc@28	292
adamc@28	293 (** Our first subgoal is [plus O O = O], which %\textit{%#<i>#is#</i>#%}% trivial by computation. *)
adamc@28	294
adamc@28	295 reflexivity.
adamc@28	296
adamc@28	297 (** Our second subgoal is more work and also demonstrates our first inductive hypothesis.
adamc@28	298
adamc@28	299 [[
adamc@28	300 n : nat
adamc@28	301 IHn : plus n O = n
adamc@28	302 ============================
adamc@28	303 plus (S n) O = S n
adamc@208	304
adamc@28	305 ]]
adamc@28	306
adam@315	307 We can start out by using computation to simplify the goal as far as we can.%\index{tactics!simpl}% *)
adamc@28	308
adamc@28	309 simpl.
adamc@28	310
adam@315	311 (** Now the conclusion is [S (][plus n O) = S n]. Using our inductive hypothesis: *)
adamc@28	312
adamc@28	313 rewrite IHn.
adamc@28	314
adamc@28	315 (** ...we get a trivial conclusion [S n = S n]. *)
adamc@28	316
adamc@28	317 reflexivity.
adamc@28	318
adam@315	319 (** Not much really went on in this proof, so the [crush] tactic from the [CpdtTactics] module can prove this theorem automatically. *)
adamc@28	320
adam@315	321 (* begin hide *)
adamc@28	322 Restart.
adam@315	323 (* end hide *)
adam@315	324 (** %\noindent \coqdockw{Restart}%#<tt>Restart</tt>#. *)
adam@315	325
adamc@28	326 induction n; crush.
adamc@28	327 Qed.
adamc@41	328 (* end thide *)
adamc@28	329
adamc@28	330 (** We can check out the induction principle at work here: *)
adamc@28	331
adamc@28	332 Check nat_ind.
adamc@208	333 (** %\vspace{-.15in}% [[
adamc@208	334 nat_ind : forall P : nat -> Prop,
adamc@208	335 P O -> (forall n : nat, P n -> P (S n)) -> forall n : nat, P n
adamc@208	336
adamc@208	337 ]]
adamc@28	338
adam@315	339 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	340
adam@315	341 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	342
adamc@28	343 Theorem S_inj : forall n m : nat, S n = S m -> n = m.
adamc@41	344 (* begin thide *)
adamc@28	345 injection 1; trivial.
adamc@28	346 Qed.
adamc@41	347 (* end thide *)
adamc@28	348
adamc@28	349 (** [injection] refers to a premise by number, adding new equalities between the corresponding arguments of equated terms that are formed with the same constructor. We end up needing to prove [n = m -> n = m], so it is unsurprising that a tactic named [trivial] is able to finish the proof.
adamc@28	350
adam@315	351 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}\textit{%#<i>#complete decision procedure for the theory of equality and uninterpreted functions#</i>#%}%, plus some smarts about inductive types.
adamc@29	352
adamc@29	353 %\medskip%
adamc@29	354
adamc@29	355 We can define a type of lists of natural numbers. *)
adamc@29	356
adamc@29	357 Inductive nat_list : Set :=
adamc@29	358 \| NNil : nat_list
adamc@29	359 \| NCons : nat -> nat_list -> nat_list.
adamc@29	360
adamc@29	361 (** Recursive definitions are straightforward extensions of what we have seen before. *)
adamc@29	362
adamc@29	363 Fixpoint nlength (ls : nat_list) : nat :=
adamc@29	364 match ls with
adamc@29	365 \| NNil => O
adamc@29	366 \| NCons _ ls' => S (nlength ls')
adamc@29	367 end.
adamc@29	368
adamc@208	369 Fixpoint napp (ls1 ls2 : nat_list) : nat_list :=
adamc@29	370 match ls1 with
adamc@29	371 \| NNil => ls2
adamc@29	372 \| NCons n ls1' => NCons n (napp ls1' ls2)
adamc@29	373 end.
adamc@29	374
adamc@29	375 (** Inductive theorem proving can again be automated quite effectively. *)
adamc@29	376
adamc@29	377 Theorem nlength_napp : forall ls1 ls2 : nat_list, nlength (napp ls1 ls2)
adamc@29	378 = plus (nlength ls1) (nlength ls2).
adamc@41	379 (* begin thide *)
adamc@29	380 induction ls1; crush.
adamc@29	381 Qed.
adamc@41	382 (* end thide *)
adamc@29	383
adamc@29	384 Check nat_list_ind.
adamc@208	385 (** %\vspace{-.15in}% [[
adamc@208	386 nat_list_ind
adamc@29	387 : forall P : nat_list -> Prop,
adamc@29	388 P NNil ->
adamc@29	389 (forall (n : nat) (n0 : nat_list), P n0 -> P (NCons n n0)) ->
adamc@29	390 forall n : nat_list, P n
adamc@29	391 ]]
adamc@29	392
adamc@29	393 %\medskip%
adamc@29	394
adam@292	395 In general, we can implement any %``%#"#tree#"#%''% types as inductive types. For example, here are binary trees of naturals. *)
adamc@29	396
adamc@29	397 Inductive nat_btree : Set :=
adamc@29	398 \| NLeaf : nat_btree
adamc@29	399 \| NNode : nat_btree -> nat -> nat_btree -> nat_btree.
adamc@29	400
adamc@29	401 Fixpoint nsize (tr : nat_btree) : nat :=
adamc@29	402 match tr with
adamc@35	403 \| NLeaf => S O
adamc@29	404 \| NNode tr1 _ tr2 => plus (nsize tr1) (nsize tr2)
adamc@29	405 end.
adamc@29	406
adamc@208	407 Fixpoint nsplice (tr1 tr2 : nat_btree) : nat_btree :=
adamc@29	408 match tr1 with
adamc@35	409 \| NLeaf => NNode tr2 O NLeaf
adamc@29	410 \| NNode tr1' n tr2' => NNode (nsplice tr1' tr2) n tr2'
adamc@29	411 end.
adamc@29	412
adamc@29	413 Theorem plus_assoc : forall n1 n2 n3 : nat, plus (plus n1 n2) n3 = plus n1 (plus n2 n3).
adamc@41	414 (* begin thide *)
adamc@29	415 induction n1; crush.
adamc@29	416 Qed.
adamc@41	417 (* end thide *)
adamc@29	418
adamc@29	419 Theorem nsize_nsplice : forall tr1 tr2 : nat_btree, nsize (nsplice tr1 tr2)
adamc@29	420 = plus (nsize tr2) (nsize tr1).
adamc@41	421 (* begin thide *)
adam@315	422 (* begin hide *)
adamc@29	423 Hint Rewrite n_plus_O plus_assoc : cpdt.
adam@315	424 (* end hide *)
adam@315	425 (** [Hint] %\coqdockw{%#<tt>#Rewrite#</tt>#%}% [n_plus_O plus_assoc : cpdt.] *)
adamc@29	426
adamc@29	427 induction tr1; crush.
adamc@29	428 Qed.
adamc@41	429 (* end thide *)
adamc@29	430
adam@315	431 (** 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	432
adamc@29	433 Check nat_btree_ind.
adamc@208	434 (** %\vspace{-.15in}% [[
adamc@208	435 nat_btree_ind
adamc@29	436 : forall P : nat_btree -> Prop,
adamc@29	437 P NLeaf ->
adamc@29	438 (forall n : nat_btree,
adamc@29	439 P n -> forall (n0 : nat) (n1 : nat_btree), P n1 -> P (NNode n n0 n1)) ->
adamc@29	440 forall n : nat_btree, P n
adam@302	441 ]]
adam@315	442
adam@315	443 We have the usual two cases, one for each constructor of [nat_btree]. *)
adamc@30	444
adamc@30	445
adamc@30	446 (** * Parameterized Types *)
adamc@30	447
adam@316	448 (** 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	449
adamc@30	450 Inductive list (T : Set) : Set :=
adamc@30	451 \| Nil : list T
adamc@30	452 \| Cons : T -> list T -> list T.
adamc@30	453
adamc@30	454 Fixpoint length T (ls : list T) : nat :=
adamc@30	455 match ls with
adamc@30	456 \| Nil => O
adamc@30	457 \| Cons _ ls' => S (length ls')
adamc@30	458 end.
adamc@30	459
adamc@208	460 Fixpoint app T (ls1 ls2 : list T) : list T :=
adamc@30	461 match ls1 with
adamc@30	462 \| Nil => ls2
adamc@30	463 \| Cons x ls1' => Cons x (app ls1' ls2)
adamc@30	464 end.
adamc@30	465
adamc@30	466 Theorem length_app : forall T (ls1 ls2 : list T), length (app ls1 ls2)
adamc@30	467 = plus (length ls1) (length ls2).
adamc@41	468 (* begin thide *)
adamc@30	469 induction ls1; crush.
adamc@30	470 Qed.
adamc@41	471 (* end thide *)
adamc@30	472
adam@316	473 (** 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}\textit{%#<i>#section#</i>#%}% mechanism. The following block of code is equivalent to the above: *)
adamc@30	474
adamc@30	475 (* begin hide *)
adamc@30	476 Reset list.
adamc@30	477 (* end hide *)
adamc@30	478
adamc@30	479 Section list.
adamc@30	480 Variable T : Set.
adamc@30	481
adamc@30	482 Inductive list : Set :=
adamc@30	483 \| Nil : list
adamc@30	484 \| Cons : T -> list -> list.
adamc@30	485
adamc@30	486 Fixpoint length (ls : list) : nat :=
adamc@30	487 match ls with
adamc@30	488 \| Nil => O
adamc@30	489 \| Cons _ ls' => S (length ls')
adamc@30	490 end.
adamc@30	491
adamc@208	492 Fixpoint app (ls1 ls2 : list) : list :=
adamc@30	493 match ls1 with
adamc@30	494 \| Nil => ls2
adamc@30	495 \| Cons x ls1' => Cons x (app ls1' ls2)
adamc@30	496 end.
adamc@30	497
adamc@30	498 Theorem length_app : forall ls1 ls2 : list, length (app ls1 ls2)
adamc@30	499 = plus (length ls1) (length ls2).
adamc@41	500 (* begin thide *)
adamc@30	501 induction ls1; crush.
adamc@30	502 Qed.
adamc@41	503 (* end thide *)
adamc@30	504 End list.
adamc@30	505
adamc@35	506 (* begin hide *)
adamc@35	507 Implicit Arguments Nil [T].
adamc@35	508 (* end hide *)
adamc@35	509
adamc@210	510 (** 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	511
adamc@202	512 Print list.
adamc@208	513 (** %\vspace{-.15in}% [[
adamc@208	514 Inductive list (T : Set) : Set :=
adam@316	515 Nil : list T \| Cons : T -> list T -> list T
adamc@208	516
adamc@202	517 ]]
adamc@30	518
adamc@202	519 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	520
adamc@30	521 Check length.
adamc@208	522 (** %\vspace{-.15in}% [[
adamc@208	523 length
adamc@30	524 : forall T : Set, list T -> nat
adamc@30	525 ]]
adamc@30	526
adamc@202	527 The parameter [T] is treated as a new argument to the induction principle, too. *)
adamc@30	528
adamc@30	529 Check list_ind.
adamc@208	530 (** %\vspace{-.15in}% [[
adamc@208	531 list_ind
adamc@30	532 : forall (T : Set) (P : list T -> Prop),
adamc@30	533 P (Nil T) ->
adamc@30	534 (forall (t : T) (l : list T), P l -> P (Cons t l)) ->
adamc@30	535 forall l : list T, P l
adamc@30	536 ]]
adamc@30	537
adamc@30	538 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	539
adamc@31	540
adamc@31	541 (** * Mutually Inductive Types *)
adamc@31	542
adamc@31	543 (** We can define inductive types that refer to each other: *)
adamc@31	544
adamc@31	545 Inductive even_list : Set :=
adamc@31	546 \| ENil : even_list
adamc@31	547 \| ECons : nat -> odd_list -> even_list
adamc@31	548
adamc@31	549 with odd_list : Set :=
adamc@31	550 \| OCons : nat -> even_list -> odd_list.
adamc@31	551
adamc@31	552 Fixpoint elength (el : even_list) : nat :=
adamc@31	553 match el with
adamc@31	554 \| ENil => O
adamc@31	555 \| ECons _ ol => S (olength ol)
adamc@31	556 end
adamc@31	557
adamc@31	558 with olength (ol : odd_list) : nat :=
adamc@31	559 match ol with
adamc@31	560 \| OCons _ el => S (elength el)
adamc@31	561 end.
adamc@31	562
adamc@208	563 Fixpoint eapp (el1 el2 : even_list) : even_list :=
adamc@31	564 match el1 with
adamc@31	565 \| ENil => el2
adamc@31	566 \| ECons n ol => ECons n (oapp ol el2)
adamc@31	567 end
adamc@31	568
adamc@208	569 with oapp (ol : odd_list) (el : even_list) : odd_list :=
adamc@31	570 match ol with
adamc@31	571 \| OCons n el' => OCons n (eapp el' el)
adamc@31	572 end.
adamc@31	573
adamc@31	574 (** 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	575
adamc@31	576 Theorem elength_eapp : forall el1 el2 : even_list,
adamc@31	577 elength (eapp el1 el2) = plus (elength el1) (elength el2).
adamc@41	578 (* begin thide *)
adamc@31	579 induction el1; crush.
adamc@31	580
adamc@31	581 (** One goal remains: [[
adamc@31	582
adamc@31	583 n : nat
adamc@31	584 o : odd_list
adamc@31	585 el2 : even_list
adamc@31	586 ============================
adamc@31	587 S (olength (oapp o el2)) = S (plus (olength o) (elength el2))
adamc@31	588 ]]
adamc@31	589
adamc@31	590 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	591
adamc@31	592 Abort.
adamc@31	593 Check even_list_ind.
adamc@208	594 (** %\vspace{-.15in}% [[
adamc@208	595 even_list_ind
adamc@31	596 : forall P : even_list -> Prop,
adamc@31	597 P ENil ->
adamc@31	598 (forall (n : nat) (o : odd_list), P (ECons n o)) ->
adamc@31	599 forall e : even_list, P e
adamc@208	600
adamc@31	601 ]]
adamc@31	602
adam@316	603 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	604
adamc@31	605 Scheme even_list_mut := Induction for even_list Sort Prop
adamc@31	606 with odd_list_mut := Induction for odd_list Sort Prop.
adamc@31	607
adam@316	608 (** 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	609
adamc@31	610 Check even_list_mut.
adamc@208	611 (** %\vspace{-.15in}% [[
adamc@208	612 even_list_mut
adamc@31	613 : forall (P : even_list -> Prop) (P0 : odd_list -> Prop),
adamc@31	614 P ENil ->
adamc@31	615 (forall (n : nat) (o : odd_list), P0 o -> P (ECons n o)) ->
adamc@31	616 (forall (n : nat) (e : even_list), P e -> P0 (OCons n e)) ->
adamc@31	617 forall e : even_list, P e
adamc@208	618
adamc@31	619 ]]
adamc@31	620
adam@316	621 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	622
adamc@31	623 Theorem n_plus_O' : forall n : nat, plus n O = n.
adamc@31	624 apply (nat_ind (fun n => plus n O = n)); crush.
adamc@31	625 Qed.
adamc@31	626
adamc@31	627 (** 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	628
adamc@31	629 This technique generalizes to our mutual example: *)
adamc@31	630
adamc@31	631 Theorem elength_eapp : forall el1 el2 : even_list,
adamc@31	632 elength (eapp el1 el2) = plus (elength el1) (elength el2).
adamc@41	633
adamc@31	634 apply (even_list_mut
adamc@31	635 (fun el1 : even_list => forall el2 : even_list,
adamc@31	636 elength (eapp el1 el2) = plus (elength el1) (elength el2))
adamc@31	637 (fun ol : odd_list => forall el : even_list,
adamc@31	638 olength (oapp ol el) = plus (olength ol) (elength el))); crush.
adamc@31	639 Qed.
adamc@41	640 (* end thide *)
adamc@31	641
adamc@31	642 (** 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	643
adamc@33	644
adamc@33	645 (** * Reflexive Types *)
adamc@33	646
adam@316	647 (** A kind of inductive type called a %\textit{%#<i>#reflexive type#</i>#%}% is defined in terms of functions that have the type being defined as their range. One very useful class of examples is in modeling variable binders. 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	648
adam@316	649 Inductive pformula : Set :=
adam@316	650 \| Truth : pformula
adam@316	651 \| Falsehood : pformula
adam@316	652 \| Conjunction : pformula -> pformula -> pformula.
adam@316	653
adam@317	654 (** A key distinction here is between, for instance, the %\emph{%#<i>#syntax#</i>#%}% [Truth] and its %\emph{%#<i>#semantics#</i>#%}% [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	655
adam@316	656 Fixpoint pformulaDenote (f : pformula) : Prop :=
adam@316	657 match f with
adam@316	658 \| Truth => True
adam@316	659 \| Falsehood => False
adam@316	660 \| Conjunction f1 f2 => pformulaDenote f1 /\ pformulaDenote f2
adam@316	661 end.
adam@316	662
adam@316	663 (** This is a 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	664
adamc@33	665 Inductive formula : Set :=
adamc@33	666 \| Eq : nat -> nat -> formula
adamc@33	667 \| And : formula -> formula -> formula
adamc@33	668 \| Forall : (nat -> formula) -> formula.
adamc@33	669
adam@316	670 (** 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	671
adamc@33	672 Example forall_refl : formula := Forall (fun x => Eq x x).
adamc@33	673
adamc@33	674 (** We can write recursive functions over reflexive types quite naturally. Here is one translating our formulas into native Coq propositions. *)
adamc@33	675
adamc@33	676 Fixpoint formulaDenote (f : formula) : Prop :=
adamc@33	677 match f with
adamc@33	678 \| Eq n1 n2 => n1 = n2
adamc@33	679 \| And f1 f2 => formulaDenote f1 /\ formulaDenote f2
adamc@33	680 \| Forall f' => forall n : nat, formulaDenote (f' n)
adamc@33	681 end.
adamc@33	682
adamc@33	683 (** We can also encode a trivial formula transformation that swaps the order of equality and conjunction operands. *)
adamc@33	684
adamc@33	685 Fixpoint swapper (f : formula) : formula :=
adamc@33	686 match f with
adamc@33	687 \| Eq n1 n2 => Eq n2 n1
adamc@33	688 \| And f1 f2 => And (swapper f2) (swapper f1)
adamc@33	689 \| Forall f' => Forall (fun n => swapper (f' n))
adamc@33	690 end.
adamc@33	691
adamc@33	692 (** It is helpful to prove that this transformation does not make true formulas false. *)
adamc@33	693
adamc@33	694 Theorem swapper_preserves_truth : forall f, formulaDenote f -> formulaDenote (swapper f).
adamc@41	695 (* begin thide *)
adamc@33	696 induction f; crush.
adamc@33	697 Qed.
adamc@41	698 (* end thide *)
adamc@33	699
adamc@33	700 (** We can take a look at the induction principle behind this proof. *)
adamc@33	701
adamc@33	702 Check formula_ind.
adamc@208	703 (** %\vspace{-.15in}% [[
adamc@208	704 formula_ind
adamc@33	705 : forall P : formula -> Prop,
adamc@33	706 (forall n n0 : nat, P (Eq n n0)) ->
adamc@33	707 (forall f0 : formula,
adamc@33	708 P f0 -> forall f1 : formula, P f1 -> P (And f0 f1)) ->
adamc@33	709 (forall f1 : nat -> formula,
adamc@33	710 (forall n : nat, P (f1 n)) -> P (Forall f1)) ->
adamc@33	711 forall f2 : formula, P f2
adamc@208	712
adamc@208	713 ]]
adamc@33	714
adamc@208	715 Focusing on the [Forall] case, which comes third, we see that we are allowed to assume that the theorem holds %\textit{%#<i>#for any application of the argument function [f1]#</i>#%}%. That is, Coq induction principles do not follow a simple rule that the textual representations of induction variables must get shorter in appeals to induction hypotheses. Luckily for us, the people behind the metatheory of Coq have verified that this flexibility does not introduce unsoundness.
adamc@33	716
adamc@33	717 %\medskip%
adamc@33	718
adam@316	719 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	720
adam@316	721 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}}\textit{%#<i>#higher-order abstract syntax (HOAS)#</i>#%}~\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: *)
adamc@33	722 (** [[
adamc@33	723 Inductive term : Set :=
adamc@33	724 \| App : term -> term -> term
adamc@33	725 \| Abs : (term -> term) -> term.
adamc@33	726 ]]
adamc@33	727
adam@316	728 <<
adam@316	729 Error: Non strictly positive occurrence of "term" in "(term -> term) -> term"
adam@316	730 >>
adam@316	731
adam@316	732 We have run afoul of the %\index{strict positivity requirement}\index{positivity requirement}\textit{%#<i>#strict positivity requirement#</i>#%}% for inductive definitions, which says that the type being defined may not occur to the left of an arrow in the type of a constructor argument. It is important that the type of a constructor is viewed in terms of a series of arguments and a result, since obviously we need recursive occurrences to the lefts of the outermost arrows if we are to have recursive occurrences at all. 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	733
adamc@33	734 Why must Coq enforce this restriction? Imagine that our last definition had been accepted, allowing us to write this function:
adamc@33	735
adamc@33	736 [[
adamc@33	737 Definition uhoh (t : term) : term :=
adamc@33	738 match t with
adamc@33	739 \| Abs f => f t
adamc@33	740 \| _ => t
adamc@33	741 end.
adamc@33	742
adamc@205	743 ]]
adamc@205	744
adamc@33	745 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	746
adam@316	747 %\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	748
adamc@33	749 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	750
adamc@34	751
adam@317	752 (** * An Interlude on Induction Principles *)
adamc@34	753
adam@317	754 (** 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	755
adam@317	756 (* begin hide *)
adamc@34	757 Print unit_ind.
adam@317	758 (* end hide *)
adam@317	759 (** %\noindent%[Print] [unit_ind.] *)
adam@317	760 (** [[
adamc@208	761 unit_ind =
adamc@208	762 fun P : unit -> Prop => unit_rect P
adamc@34	763 : forall P : unit -> Prop, P tt -> forall u : unit, P u
adamc@208	764
adamc@34	765 ]]
adamc@34	766
adam@317	767 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	768
adamc@34	769 Check unit_rect.
adamc@208	770 (** %\vspace{-.15in}% [[
adamc@208	771 unit_rect
adamc@34	772 : forall P : unit -> Type, P tt -> forall u : unit, P u
adamc@208	773
adamc@34	774 ]]
adamc@34	775
adamc@34	776 [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	777
adam@317	778 (* begin hide *)
adamc@34	779 Print unit_rec.
adam@317	780 (* end hide *)
adam@317	781 (** %\noindent%[Print] [unit_rec.] *)
adam@317	782 (** [[
adamc@208	783 unit_rec =
adamc@208	784 fun P : unit -> Set => unit_rect P
adamc@34	785 : forall P : unit -> Set, P tt -> forall u : unit, P u
adamc@208	786
adamc@34	787 ]]
adamc@34	788
adam@317	789 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	790
adamc@34	791 Definition always_O (u : unit) : nat :=
adamc@34	792 match u with
adamc@34	793 \| tt => O
adamc@34	794 end.
adamc@34	795
adamc@34	796 Definition always_O' (u : unit) : nat :=
adamc@34	797 unit_rec (fun _ : unit => nat) O u.
adamc@34	798
adamc@34	799 (** 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	800
adam@317	801 (* begin hide *)
adamc@34	802 Print unit_rect.
adam@317	803 (* end hide *)
adam@317	804 (** %\noindent%[Print] [unit_rect.] *)
adam@317	805 (** [[
adamc@208	806 unit_rect =
adamc@208	807 fun (P : unit -> Type) (f : P tt) (u : unit) =>
adamc@208	808 match u as u0 return (P u0) with
adamc@208	809 \| tt => f
adamc@208	810 end
adamc@34	811 : forall P : unit -> Type, P tt -> forall u : unit, P u
adamc@208	812
adamc@34	813 ]]
adamc@34	814
adam@317	815 The only new wrinkle here is the annotations on the [match] expression. This is a %\index{dependent pattern matching}\emph{%#<i>#dependently typed#</i>#%}% pattern match, because the %\emph{%#<i>#type#</i>#%}% of the expression depends on the %\emph{%#<i>#value#</i>#%}% 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	816
adam@317	817 %\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	818
adamc@34	819 To prove that [unit_rect] is nothing special, we can reimplement it manually. *)
adamc@34	820
adamc@34	821 Definition unit_rect' (P : unit -> Type) (f : P tt) (u : unit) :=
adamc@208	822 match u with
adamc@34	823 \| tt => f
adamc@34	824 end.
adamc@34	825
adam@317	826 (** We rely on Coq's heuristics for inferring [match] annotations, which are not consulted in the pretty-printing of terms.
adamc@34	827
adam@317	828 We can check the implementation of %\coqdocdefinition{%#<tt>#nat_rect#</tt>#%}% as well: *)
adamc@34	829
adam@317	830 (* begin hide *)
adamc@34	831 Print nat_rect.
adam@317	832 (* end hide *)
adam@317	833 (** %\noindent%[Print] %\coqdocdefinition{%#<tt>#nat_rect#</tt>#%}%[.] *)
adam@317	834
adam@317	835 (** %\hspace{-.05in}\coqdocdefinition{%#<tt>#nat_rect#</tt>#%}% [=] *)
adam@317	836 (** %\vspace{-.05in}% [[
adamc@208	837 fun (P : nat -> Type) (f : P O) (f0 : forall n : nat, P n -> P (S n)) =>
adamc@208	838 fix F (n : nat) : P n :=
adamc@208	839 match n as n0 return (P n0) with
adamc@208	840 \| O => f
adamc@208	841 \| S n0 => f0 n0 (F n0)
adamc@208	842 end
adamc@208	843 : forall P : nat -> Type,
adamc@208	844 P O -> (forall n : nat, P n -> P (S n)) -> forall n : nat, P n
adamc@208	845 ]]
adamc@34	846
adam@317	847 Now we have an actual recursive definition. %\index{Gallina terms!fix}%[fix] expressions are an anonymous form of [Fixpoint], just as [fun] expressions stand for anonymous non-recursive functions. Beyond that, the syntax of [fix] mirrors that of [Fixpoint]. We can understand the definition of %\coqdocdefinition{%#<tt>#nat_rect#</tt>#%}% better by reimplementing [nat_ind] using sections. *)
adamc@34	848
adam@317	849 Section nat_ind'.
adamc@208	850 (** First, we have the property of natural numbers that we aim to prove. *)
adamc@34	851
adam@317	852 Variable P : nat -> Prop.
adamc@34	853
adam@317	854 (** 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	855
adam@317	856 Hypothesis O_case : P O.
adamc@34	857
adamc@208	858 (** Next is a proof of the [S] case, which may assume an inductive hypothesis. *)
adamc@34	859
adam@317	860 Hypothesis S_case : forall n : nat, P n -> P (S n).
adamc@34	861
adamc@208	862 (** Finally, we define a recursive function to tie the pieces together. *)
adamc@34	863
adam@317	864 Fixpoint nat_ind' (n : nat) : P n :=
adam@317	865 match n with
adam@317	866 \| O => O_case
adam@317	867 \| S n' => S_case (nat_ind' n')
adam@317	868 end.
adam@317	869 End nat_ind'.
adamc@34	870
adam@317	871 (** 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 %\coqdocdefinition{%#<tt>#nat_rect#</tt>#%}%.
adamc@34	872
adam@317	873 %\medskip%
adamc@34	874
adam@317	875 We can also examine the definition of [even_list_mut], which we generated with [Scheme] for a mutually recursive type. *)
adamc@34	876
adam@317	877 (* begin hide *)
adam@317	878 Print even_list_mut.
adam@317	879 (* end hide *)
adam@317	880 (** %\noindent%[Print] %\coqdocdefinition{%#<tt>#even_list_mut#</tt>#%}%[.] *)
adam@317	881 (** [[
adam@317	882 even_list_mut =
adam@317	883 fun (P : even_list -> Prop) (P0 : odd_list -> Prop)
adam@317	884 (f : P ENil) (f0 : forall (n : nat) (o : odd_list), P0 o -> P (ECons n o))
adam@317	885 (f1 : forall (n : nat) (e : even_list), P e -> P0 (OCons n e)) =>
adam@317	886 fix F (e : even_list) : P e :=
adam@317	887 match e as e0 return (P e0) with
adam@317	888 \| ENil => f
adam@317	889 \| ECons n o => f0 n o (F0 o)
adam@317	890 end
adam@317	891 with F0 (o : odd_list) : P0 o :=
adam@317	892 match o as o0 return (P0 o0) with
adam@317	893 \| OCons n e => f1 n e (F e)
adam@317	894 end
adam@317	895 for F
adam@317	896 : forall (P : even_list -> Prop) (P0 : odd_list -> Prop),
adam@317	897 P ENil ->
adam@317	898 (forall (n : nat) (o : odd_list), P0 o -> P (ECons n o)) ->
adam@317	899 (forall (n : nat) (e : even_list), P e -> P0 (OCons n e)) ->
adam@317	900 forall e : even_list, P e
adamc@34	901
adam@317	902 ]]
adamc@34	903
adam@317	904 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	905
adam@317	906 Section even_list_mut'.
adam@317	907 (** First, we need the properties that we are proving. *)
adamc@208	908
adam@317	909 Variable Peven : even_list -> Prop.
adam@317	910 Variable Podd : odd_list -> Prop.
adamc@208	911
adam@317	912 (** Next, we need proofs of the three cases. *)
adamc@208	913
adam@317	914 Hypothesis ENil_case : Peven ENil.
adam@317	915 Hypothesis ECons_case : forall (n : nat) (o : odd_list), Podd o -> Peven (ECons n o).
adam@317	916 Hypothesis OCons_case : forall (n : nat) (e : even_list), Peven e -> Podd (OCons n e).
adamc@208	917
adam@317	918 (** Finally, we define the recursive functions. *)
adamc@208	919
adam@317	920 Fixpoint even_list_mut' (e : even_list) : Peven e :=
adam@317	921 match e with
adam@317	922 \| ENil => ENil_case
adam@317	923 \| ECons n o => ECons_case n (odd_list_mut' o)
adam@317	924 end
adam@317	925 with odd_list_mut' (o : odd_list) : Podd o :=
adam@317	926 match o with
adam@317	927 \| OCons n e => OCons_case n (even_list_mut' e)
adam@317	928 end.
adamc@34	929 End even_list_mut'.
adamc@34	930
adamc@34	931 (** 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	932
adamc@34	933 Section formula_ind'.
adamc@34	934 Variable P : formula -> Prop.
adamc@38	935 Hypothesis Eq_case : forall n1 n2 : nat, P (Eq n1 n2).
adamc@38	936 Hypothesis And_case : forall f1 f2 : formula,
adamc@34	937 P f1 -> P f2 -> P (And f1 f2).
adamc@38	938 Hypothesis Forall_case : forall f : nat -> formula,
adamc@34	939 (forall n : nat, P (f n)) -> P (Forall f).
adamc@34	940
adamc@34	941 Fixpoint formula_ind' (f : formula) : P f :=
adamc@208	942 match f with
adamc@34	943 \| Eq n1 n2 => Eq_case n1 n2
adamc@34	944 \| And f1 f2 => And_case (formula_ind' f1) (formula_ind' f2)
adamc@34	945 \| Forall f' => Forall_case f' (fun n => formula_ind' (f' n))
adamc@34	946 end.
adamc@34	947 End formula_ind'.
adamc@34	948
adam@317	949 (** It is apparent that induction principle implementations involve some tedium but not terribly much creativity. *)
adam@317	950
adamc@35	951
adamc@35	952 (** * Nested Inductive Types *)
adamc@35	953
adamc@35	954 (** 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	955
adamc@35	956 Inductive nat_tree : Set :=
adamc@35	957 \| NLeaf' : nat_tree
adamc@35	958 \| NNode' : nat -> list nat_tree -> nat_tree.
adamc@35	959
adam@317	960 (** This is an example of a %\index{nested inductive type}\textit{%#<i>#nested#</i>#%}% inductive type definition, because we use the type we are defining as an argument to a parametrized type family. Coq will not allow all such definitions; it effectively pretends that we are defining [nat_tree] mutually with a version of [list] specialized to [nat_tree], checking that the resulting expanded definition satisfies the usual rules. For instance, if we replaced [list] with a type family that used its parameter as a function argument, then the definition would be rejected as violating the positivity restriction.
adamc@35	961
adam@317	962 Like we encountered for mutual inductive types, we find that the automatically generated induction principle for [nat_tree] is too weak. *)
adamc@35	963
adamc@35	964 Check nat_tree_ind.
adamc@208	965 (** %\vspace{-.15in}% [[
adamc@208	966 nat_tree_ind
adamc@35	967 : forall P : nat_tree -> Prop,
adamc@35	968 P NLeaf' ->
adamc@35	969 (forall (n : nat) (l : list nat_tree), P (NNode' n l)) ->
adamc@35	970 forall n : nat_tree, P n
adamc@208	971
adamc@35	972 ]]
adamc@35	973
adam@317	974 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	975
adamc@35	976 First, we will need an auxiliary definition, characterizing what it means for a property to hold of every element of a list. *)
adamc@35	977
adamc@35	978 Section All.
adamc@35	979 Variable T : Set.
adamc@35	980 Variable P : T -> Prop.
adamc@35	981
adamc@35	982 Fixpoint All (ls : list T) : Prop :=
adamc@35	983 match ls with
adamc@35	984 \| Nil => True
adamc@35	985 \| Cons h t => P h /\ All t
adamc@35	986 end.
adamc@35	987 End All.
adamc@35	988
adam@317	989 (** It will be useful to review the definitions of [True] and [/\], since we will want to write manual proofs of them below. *)
adamc@35	990
adam@317	991 (* begin hide *)
adamc@35	992 Print True.
adam@317	993 (* end hide *)
adam@317	994 (** %\noindent%[Print] %\coqdocinductive{%#<tt>#True#</tt>#%}%[.] *)
adam@317	995 (** [[
adamc@208	996 Inductive True : Prop := I : True
adamc@208	997 ]]
adamc@35	998
adamc@35	999 That is, [True] is a proposition with exactly one proof, [I], which we may always supply trivially.
adamc@35	1000
adam@317	1001 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}\coqdockw{%#<tt>#Locate#</tt>#%}% command, whose argument may be a parsing token.%\index{Gallina terms!and}% *)
adamc@35	1002
adam@317	1003 (* begin hide *)
adamc@35	1004 Locate "/\".
adam@317	1005 (* end hide *)
adam@317	1006 (** %\noindent \coqdockw{Locate}%#<tt>Locate</tt># ["/\".] *)
adam@317	1007 (** [[
adam@317	1008 "A /\ B" := and A B : type_scope (default interpretation)
adam@302	1009 ]]
adam@302	1010 *)
adamc@35	1011
adam@317	1012 (* begin hide *)
adamc@35	1013 Print and.
adam@317	1014 (* end hide *)
adam@317	1015 (** %\noindent%[Print] %\coqdocinductive{%#<tt>#and#</tt>#%}%[.] *)
adam@317	1016 (** [[
adamc@208	1017 Inductive and (A : Prop) (B : Prop) : Prop := conj : A -> B -> A /\ B
adam@317	1018 ]]
adam@317	1019 %\vspace{-.1in}%
adam@317	1020 <<
adamc@208	1021 For conj: Arguments A, B are implicit
adam@317	1022 >>
adamc@35	1023
adam@317	1024 In addition to the definition of %\coqdocinductive{%#<tt>#and#</tt>#%}% 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	1025
adamc@35	1026 %\medskip%
adamc@35	1027
adamc@35	1028 Now we create a section for our induction principle, following the same basic plan as in the last section of this chapter. *)
adamc@35	1029
adamc@35	1030 Section nat_tree_ind'.
adamc@35	1031 Variable P : nat_tree -> Prop.
adamc@35	1032
adamc@38	1033 Hypothesis NLeaf'_case : P NLeaf'.
adamc@38	1034 Hypothesis NNode'_case : forall (n : nat) (ls : list nat_tree),
adamc@35	1035 All P ls -> P (NNode' n ls).
adamc@35	1036
adamc@35	1037 (** 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	1038
adamc@35	1039 [[
adamc@35	1040 Fixpoint nat_tree_ind' (tr : nat_tree) : P tr :=
adamc@208	1041 match tr with
adamc@35	1042 \| NLeaf' => NLeaf'_case
adamc@35	1043 \| NNode' n ls => NNode'_case n ls (list_nat_tree_ind ls)
adamc@35	1044 end
adamc@35	1045
adamc@35	1046 with 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.
adamc@35	1051
adamc@205	1052 ]]
adamc@205	1053
adam@317	1054 Coq rejects this definition, saying
adam@317	1055 <<
adam@317	1056 Recursive call to nat_tree_ind' has principal argument equal to "tr"
adam@317	1057 instead of rest.
adam@317	1058 >>
adam@317	1059
adam@317	1060 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	1061
adamc@35	1062 Fixpoint nat_tree_ind' (tr : nat_tree) : P tr :=
adamc@208	1063 match tr with
adamc@35	1064 \| NLeaf' => NLeaf'_case
adamc@35	1065 \| NNode' n ls => NNode'_case n ls
adamc@35	1066 ((fix list_nat_tree_ind (ls : list nat_tree) : All P ls :=
adamc@208	1067 match ls with
adamc@35	1068 \| Nil => I
adamc@35	1069 \| Cons tr rest => conj (nat_tree_ind' tr) (list_nat_tree_ind rest)
adamc@35	1070 end) ls)
adamc@35	1071 end.
adamc@35	1072
adam@279	1073 (** We include an anonymous [fix] version of [list_nat_tree_ind] that is literally %\textit{%#<i>#nested#</i>#%}% inside the definition of the recursive function corresponding to the inductive definition that had the nested use of [list]. *)
adamc@35	1074
adamc@35	1075 End nat_tree_ind'.
adamc@35	1076
adamc@35	1077 (** 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	1078
adamc@35	1079 Section map.
adamc@35	1080 Variables T T' : Set.
adam@317	1081 Variable F : T -> T'.
adamc@35	1082
adamc@35	1083 Fixpoint map (ls : list T) : list T' :=
adamc@35	1084 match ls with
adamc@35	1085 \| Nil => Nil
adam@317	1086 \| Cons h t => Cons (F h) (map t)
adamc@35	1087 end.
adamc@35	1088 End map.
adamc@35	1089
adamc@35	1090 Fixpoint sum (ls : list nat) : nat :=
adamc@35	1091 match ls with
adamc@35	1092 \| Nil => O
adamc@35	1093 \| Cons h t => plus h (sum t)
adamc@35	1094 end.
adamc@35	1095
adamc@35	1096 (** Now we can define a size function over our trees. *)
adamc@35	1097
adamc@35	1098 Fixpoint ntsize (tr : nat_tree) : nat :=
adamc@35	1099 match tr with
adamc@35	1100 \| NLeaf' => S O
adamc@35	1101 \| NNode' _ trs => S (sum (map ntsize trs))
adamc@35	1102 end.
adamc@35	1103
adamc@35	1104 (** 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	1105
adamc@208	1106 Fixpoint ntsplice (tr1 tr2 : nat_tree) : nat_tree :=
adamc@35	1107 match tr1 with
adamc@35	1108 \| NLeaf' => NNode' O (Cons tr2 Nil)
adamc@35	1109 \| NNode' n Nil => NNode' n (Cons tr2 Nil)
adamc@35	1110 \| NNode' n (Cons tr trs) => NNode' n (Cons (ntsplice tr tr2) trs)
adamc@35	1111 end.
adamc@35	1112
adamc@35	1113 (** 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	1114
adamc@41	1115 (* begin thide *)
adamc@35	1116 Lemma plus_S : forall n1 n2 : nat,
adamc@35	1117 plus n1 (S n2) = S (plus n1 n2).
adamc@35	1118 induction n1; crush.
adamc@35	1119 Qed.
adamc@41	1120 (* end thide *)
adamc@35	1121
adamc@35	1122 (** Now we begin the proof of the theorem, adding the lemma [plus_S] as a hint. *)
adamc@35	1123
adamc@35	1124 Theorem ntsize_ntsplice : forall tr1 tr2 : nat_tree, ntsize (ntsplice tr1 tr2)
adamc@35	1125 = plus (ntsize tr2) (ntsize tr1).
adamc@41	1126 (* begin thide *)
adam@317	1127 (* begin hide *)
adamc@35	1128 Hint Rewrite plus_S : cpdt.
adam@317	1129 (* end hide *)
adam@317	1130 (** [Hint] %\coqdockw{%#<tt>#Rewrite#</tt>#%}% [plus_S : cpdt.] *)
adamc@35	1131
adam@317	1132 (** 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	1133
adamc@35	1134 induction tr1 using nat_tree_ind'; crush.
adamc@35	1135
adamc@35	1136 (** One subgoal remains: [[
adamc@35	1137 n : nat
adamc@35	1138 ls : list nat_tree
adamc@35	1139 H : All
adamc@35	1140 (fun tr1 : nat_tree =>
adamc@35	1141 forall tr2 : nat_tree,
adamc@35	1142 ntsize (ntsplice tr1 tr2) = plus (ntsize tr2) (ntsize tr1)) ls
adamc@35	1143 tr2 : nat_tree
adamc@35	1144 ============================
adamc@35	1145 ntsize
adamc@35	1146 match ls with
adamc@35	1147 \| Nil => NNode' n (Cons tr2 Nil)
adamc@35	1148 \| Cons tr trs => NNode' n (Cons (ntsplice tr tr2) trs)
adamc@35	1149 end = S (plus (ntsize tr2) (sum (map ntsize ls)))
adamc@208	1150
adamc@35	1151 ]]
adamc@35	1152
adamc@35	1153 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	1154
adamc@35	1155 destruct ls; crush.
adamc@35	1156
adam@317	1157 (** We can go further in automating the proof by exploiting the hint mechanism.%\index{Vernacular commands!Hint Extern}% *)
adamc@35	1158
adam@317	1159 (* begin hide *)
adamc@35	1160 Restart.
adam@317	1161 (* end hide *)
adam@317	1162 (** %\hspace{-.075in}\coqdockw{%#<tt>#Restart#</tt>#%}%[.] *)
adam@317	1163
adamc@35	1164 Hint Extern 1 (ntsize (match ?LS with Nil => _ \| Cons _ _ => _ end) = _) =>
adamc@35	1165 destruct LS; crush.
adamc@35	1166 induction tr1 using nat_tree_ind'; crush.
adamc@35	1167 Qed.
adamc@41	1168 (* end thide *)
adamc@35	1169
adamc@35	1170 (** 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	1171
adam@317	1172 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	1173
adamc@36	1174
adamc@36	1175 (** * Manual Proofs About Constructors *)
adamc@36	1176
adam@317	1177 (** 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	1178
adamc@36	1179 Theorem true_neq_false : true <> false.
adamc@208	1180
adamc@41	1181 (* begin thide *)
adam@317	1182 (** 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	1183
adamc@36	1184 red.
adamc@36	1185 (** [[
adamc@36	1186 ============================
adamc@36	1187 true = false -> False
adamc@208	1188
adamc@36	1189 ]]
adamc@36	1190
adam@317	1191 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	1192
adamc@36	1193 intro H.
adamc@36	1194 (** [[
adamc@36	1195 H : true = false
adamc@36	1196 ============================
adamc@36	1197 False
adamc@208	1198
adamc@36	1199 ]]
adamc@36	1200
adamc@36	1201 This is the point in the proof where we apply some creativity. We define a function whose utility will become clear soon. *)
adamc@36	1202
adam@317	1203 Definition toProp (b : bool) := if b then True else False.
adamc@36	1204
adam@317	1205 (** 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}\coqdockw{%#<tt>#change#</tt>#%}% tactic will let us change the conclusion to [toProp false]. The general form %\coqdockw{%#<tt>#change#</tt>#%}% [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	1206
adam@317	1207 (* begin hide *)
adam@317	1208 change (toProp false).
adam@317	1209 (* end hide *)
adam@317	1210 (** %\hspace{-.075in}\coqdockw{%#<tt>#change#</tt>#%}% [(][toProp false).] *)
adamc@36	1211 (** [[
adamc@36	1212 H : true = false
adamc@36	1213 ============================
adam@317	1214 toProp false
adamc@208	1215
adamc@36	1216 ]]
adamc@36	1217
adam@317	1218 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	1219
adamc@36	1220 rewrite <- H.
adamc@36	1221 (** [[
adamc@36	1222 H : true = false
adamc@36	1223 ============================
adam@317	1224 toProp true
adamc@208	1225
adamc@36	1226 ]]
adamc@36	1227
adamc@36	1228 We are almost done. Just how close we are to done is revealed by computational simplification. *)
adamc@36	1229
adamc@36	1230 simpl.
adamc@36	1231 (** [[
adamc@36	1232 H : true = false
adamc@36	1233 ============================
adamc@36	1234 True
adamc@208	1235
adam@302	1236 ]]
adam@302	1237 *)
adamc@36	1238
adamc@36	1239 trivial.
adamc@36	1240 Qed.
adamc@41	1241 (* end thide *)
adamc@36	1242
adamc@36	1243 (** I have no trivial automated version of this proof to suggest, beyond using [discriminate] or [congruence] in the first place.
adamc@36	1244
adamc@36	1245 %\medskip%
adamc@36	1246
adamc@36	1247 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	1248
adamc@36	1249 Theorem S_inj' : forall n m : nat, S n = S m -> n = m.
adamc@41	1250 (* begin thide *)
adamc@36	1251 intros n m H.
adam@317	1252 (* begin hide *)
adamc@36	1253 change (pred (S n) = pred (S m)).
adam@317	1254 (* end hide *)
adam@317	1255 (** %\hspace{-.075in}\coqdockw{%#<tt>#change#</tt>#%}% [(][pred (][S n) = pred (][S m)).] *)
adam@317	1256
adamc@36	1257 rewrite H.
adamc@36	1258 reflexivity.
adamc@36	1259 Qed.
adamc@41	1260 (* end thide *)
adamc@36	1261
adam@317	1262 (** The key piece of creativity in this theorem comes in the use of the natural number predecessor function [pred]. Embodied in the implementation of %\coqdockw{%#<tt>#injectivity#</tt>#%}% is a generic recipe for writing such type-specific functions.
adam@317	1263
adam@317	1264 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. *)
adam@317	1265
adamc@37	1266
adamc@37	1267 (** * Exercises *)
adamc@37	1268
adamc@37	1269 (** %\begin{enumerate}%#<ol>#
adamc@37	1270
adam@292	1271 %\item%#<li># Define an inductive type [truth] with three constructors, [Yes], [No], and [Maybe]. [Yes] stands for certain truth, [No] for certain falsehood, and [Maybe] for an unknown situation. Define %``%#"#not,#"#%''% %``%#"#and,#"#%''% and %``%#"#or#"#%''% for this replacement boolean algebra. Prove that your implementation of %``%#"#and#"#%''% is commutative and distributes over your implementation of %``%#"#or.#"#%''%#</li>#
adamc@37	1272
adam@318	1273 %\item%#<li># Define an inductive type [slist] that implements lists with support for constant-time concatenation. This type should be polymorphic in a choice of type for data values in lists. The type [slist] should have three constructors, for empty lists, singleton lists, and concatenation. Define a function [flatten] that converts [slist]s to [list]s. (You will want to run [Require Import] %\coqdocconstructor{%#<tt>#List#</tt>#%}%[.] to bring list definitions into scope.) Finally, prove that [flatten] distributes over concatenation, where the two sides of your quantified equality will use the [slist] and [list] versions of concatenation, as appropriate. Recall from Chapter 2 that the infix operator [++] is syntactic sugar for the [list] concatenation function [app].#</li>#
adamc@38	1274
adam@318	1275 %\item%#<li># $\star$ Modify the first example language of Chapter 2 to include variables, where variables are represented with [nat]. Extend the syntax and semantics of expressions to accommodate the change. Your new [expDenote] function should take as a new extra first argument a value of type [var -> nat], where [var] is a synonym for naturals-as-variables, and the function assigns a value to each variable. Define a constant folding function which does a bottom-up pass over an expression, at each stage replacing every binary operation on constants with an equivalent constant. Prove that constant folding preserves the meanings of expressions.#</li>#
adam@318	1276
adam@318	1277 %\item%#<li># $\star$ Reimplement the second example language of Chapter 2 to use mutually inductive types instead of dependent types. That is, define two separate (non-dependent) inductive types [nat_exp] and [bool_exp] for expressions of the two different types, rather than a single indexed type. To keep things simple, you may consider only the binary operators that take naturals as operands. Add natural number variables to the language, as in the last exercise, and add an %``%#"#if#"#%''% expression form taking as arguments one boolean expression and two natural number expressions. Define semantics and constant-folding functions for this new language. Your constant folding should simplify not just binary operations (returning naturals or booleans) with known arguments, but also %``%#"#if#"#%''% expressions with known values for their test expressions but possibly undetermined %``%#"#then#"#%''% and %``%#"#else#"#%''% cases. Prove that constant-folding a natural number expression preserves its meaning.#</li>#
adam@318	1278
adam@318	1279 %\item%#<li># Define mutually inductive types of even and odd natural numbers, such that any natural number is isomorphic to a value of one of the two types. (This problem does not ask you to prove that correspondence, though some interpretations of the task may be interesting exercises.) Write a function that computes the sum of two even numbers, such that the function type guarantees that the output is even as well. Prove that this function is commutative.#</li>#
adamc@38	1280
adamc@38	1281 %\item%#<li># Using a reflexive inductive definition, define a type [nat_tree] of infinitary trees, with natural numbers at their leaves and a countable infinity of new trees branching out of each internal node. Define a function [increment] that increments the number in every leaf of a [nat_tree]. Define a function [leapfrog] over a natural [i] and a tree [nt]. [leapfrog] should recurse into the [i]th child of [nt], the [i+1]st child of that node, the [i+2]nd child of the next node, and so on, until reaching a leaf, in which case [leapfrog] should return the number at that leaf. Prove that the result of any call to [leapfrog] is incremented by one by calling [increment] on the tree.#</li>#
adamc@38	1282
adamc@38	1283 %\item%#<li># Define a type of trees of trees of trees of (repeat to infinity). That is, define an inductive type [trexp], whose members are either base cases containing natural numbers or binary trees of [trexp]s. Base your definition on a parameterized binary tree type [btree] that you will also define, so that [trexp] is defined as a nested inductive type. Define a function [total] that sums all of the naturals at the leaves of a [trexp]. Define a function [increment] that increments every leaf of a [trexp] by one. Prove that, for all [tr], [total (increment tr) >= total tr]. On the way to finishing this proof, you will probably want to prove a lemma and add it as a hint using the syntax [Hint Resolve name_of_lemma.].#</li>#
adamc@38	1284
adam@318	1285 %\item%#<li># $\star$ Prove discrimination and injectivity theorems for the [nat_btree] type defined earlier in this chapter. In particular, without using the tactics [discriminate], [injection], or [congruence], prove that no leaf equals any node, and prove that two equal nodes carry the same natural number.#</li>#
adamc@37	1286
adamc@37	1287 #</ol>#%\end{enumerate}% *)

Mercurial > cpdt > repo

annotate src/InductiveTypes.v @ 321:a9edc6ecd904