### annotate src/InductiveTypes.v @ 315:72bffb046797

Pass through InductiveTypes, through end of recursive types
author Adam Chlipala Sun, 11 Sep 2011 16:26:48 -0400 d5787b70cf48 2aaff91f5258
rev   line source
adamc@26 10 (* begin hide *)
adamc@26 16 (* end hide *)
adamc@74 19 (** %\part{Basic Programming and Proving}
adamc@74 21 \chapter{Introducing Inductive Types}% *)
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 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 28 (** * Proof Terms *)
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 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 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 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 38 Check (fun x : nat => x).
adam@315 39 (** [: nat -> nat] *)
adam@315 41 (** Consider this alternate program, which is almost identical to the last one. *)
adam@315 43 Check (fun x : True => x).
adam@315 44 (** [: True -> True] *)
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 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 51 (** [: True] *)
adam@315 53 (** With [I], we can prove another simple propositional theorem. *)
adam@315 55 Check (fun _ : False => I).
adam@315 56 (** [: False -> True] *)
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 60 Check (fun x : False => x).
adam@315 61 (** [: False -> False] *)
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 65 Check (fun x : False => match x with end : True).
adam@315 66 (** [: False -> True] *)
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 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). *)
adamc@26 73 (** * Enumerations *)
adam@315 75 (** 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.
adam@315 77 The singleton type [unit] is an inductive type:%\index{Gallina terms!unit}\index{Gallina terms!tt}% *)
adamc@26 79 Inductive unit : Set :=
adamc@26 82 (** 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@208 85 (** [unit : Set] *)
adamc@208 88 (** [tt : unit] *)
adamc@26 90 (** We can prove that [unit] is a genuine singleton type. *)
adamc@26 92 Theorem unit_singleton : forall x : unit, x = tt.
adam@315 94 (** 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@41 96 (* begin thide *)
adamc@208 99 (** The goal changes to:
adamc@26 105 (** ...which we can discharge trivially. *)
adamc@41 109 (* end thide *)
adam@315 111 (** 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}% [[
adam@292 117 %\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.
adam@315 119 What exactly %\textit{%#<i>#is#</i>#%}% the %\index{induction principles}%induction principle for [unit]? We can ask Coq: *)
adamc@208 122 (** [unit_ind : forall P : unit -> Prop, P tt -> forall u : unit, P u] *)
adam@315 124 (** 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.
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.
adam@315 128 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@315 130 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.
adam@315 134 We can define an inductive type even simpler than [unit]:%\index{Gallina terms!Empty\_set}% *)
adamc@26 136 Inductive Empty_set : Set := .
adamc@26 138 (** [Empty_set] has no elements. We can prove fun theorems about it: *)
adamc@26 140 Theorem the_sky_is_falling : forall x : Empty_set, 2 + 2 = 5.
adamc@41 141 (* begin thide *)
adamc@41 144 (* end thide *)
adamc@32 146 (** 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. 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@26 148 We can see the induction principle that made this proof so easy: *)
adam@315 151 (** [Empty_set_ind : forall (][P : Empty_set -> Prop) (e : Empty_set), P e] *)
adam@315 153 (** 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@26 155 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@26 157 Definition e2u (e : Empty_set) : unit := match e with end.
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.
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@26 165 Inductive bool : Set :=
adamc@26 169 (** We can use less vacuous pattern matching to define boolean negation. *)
adam@279 171 Definition negb (b : bool) : bool :=
adamc@26 173 | true => false
adamc@26 174 | false => true
adamc@27 177 (** An alternative definition desugars to the above: *)
adam@279 179 Definition negb' (b : bool) : bool :=
adamc@27 180 if b then false else true.
adam@279 182 (** We might want to prove that [negb] is its own inverse operation. *)
adam@279 184 Theorem negb_inverse : forall b : bool, negb (negb b) = b.
adamc@41 185 (* begin thide *)
adamc@208 188 (** After we case-analyze on [b], we are left with one subgoal for each constructor of [bool].
adam@315 190 %\vspace{.1in} \noindent 2 \coqdockw{subgoals}\vspace{-.1in}%#<tt>2 subgoals</tt>#
adam@279 194 negb (negb true) = true
adam@315 196 %\noindent \coqdockw{subgoal} 2 \coqdockw{is}:%#<tt>subgoal 2 is</tt>#
adam@279 198 negb (negb false) = false
adamc@26 202 The first subgoal follows by Coq's rules of computation, so we can dispatch it easily: *)
adam@315 206 (** Likewise for the second subgoal, so we can restart the proof and give a very compact justification.%\index{Vernacular commands!Restart}% *)
adam@315 208 (* begin hide *)
adam@315 210 (* end hide *)
adam@315 211 (** %\noindent \coqdockw{Restart}%#<tt>Restart</tt>#. *)
adamc@41 215 (* end thide *)
adam@315 217 (** Another theorem about booleans illustrates another useful tactic.%\index{tactics!discriminate}% *)
adam@279 219 Theorem negb_ineq : forall b : bool, negb b <> b.
adamc@41 220 (* begin thide *)
adamc@41 223 (* end thide *)
adamc@27 225 (** [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@27 227 At this point, it is probably not hard to guess what the underlying induction principle for [bool] is. *)
adamc@208 230 (** [bool_ind : forall P : bool -> Prop, P true -> P false -> forall b : bool, P b] *)
adam@315 232 (** That is, to prove that a property describes all [bool]s, prove that it describes both [true] and [false].
adam@315 234 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. *)
adamc@28 237 (** * Simple Recursive Types *)
adam@315 239 (** 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@28 241 Inductive nat : Set :=
adamc@28 242 | O : nat
adamc@28 243 | S : nat -> nat.
adam@315 245 (** [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@28 247 Pattern matching works as we demonstrated in the last chapter: *)
adamc@28 249 Definition isZero (n : nat) : bool :=
adamc@28 251 | O => true
adamc@28 252 | S _ => false
adamc@28 255 Definition pred (n : nat) : nat :=
adamc@28 257 | O => O
adamc@28 258 | S n' => n'
adamc@28 261 (** We can prove theorems by case analysis: *)
adamc@28 263 Theorem S_isZero : forall n : nat, isZero (pred (S (S n))) = false.
adamc@41 264 (* begin thide *)
adamc@41 267 (* end thide *)
adamc@28 269 (** We can also now get into genuine inductive theorems. First, we will need a recursive function, to make things interesting. *)
adamc@208 271 Fixpoint plus (n m : nat) : nat :=
adamc@28 273 | O => m
adamc@28 274 | S n' => S (plus n' m)
adamc@208 277 (** Recall that [Fixpoint] is Coq's mechanism for recursive function definitions. Some theorems about [plus] can be proved without induction. *)
adamc@28 279 Theorem O_plus_n : forall n : nat, plus O n = n.
adamc@41 280 (* begin thide *)
adamc@41 283 (* end thide *)
adamc@208 285 (** 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@28 287 Theorem n_plus_O : forall n : nat, plus n O = n.
adamc@41 288 (* begin thide *)
adamc@28 291 (** Our first subgoal is [plus O O = O], which %\textit{%#<i>#is#</i>#%}% trivial by computation. *)
adamc@28 295 (** Our second subgoal is more work and also demonstrates our first inductive hypothesis.
adamc@28 299 IHn : plus n O = n
adamc@28 301 plus (S n) O = S n
adam@315 305 We can start out by using computation to simplify the goal as far as we can.%\index{tactics!simpl}% *)
adam@315 309 (** Now the conclusion is [S (][plus n O) = S n]. Using our inductive hypothesis: *)
adamc@28 313 (** ...we get a trivial conclusion [S n = S n]. *)
adam@315 317 (** Not much really went on in this proof, so the [crush] tactic from the [CpdtTactics] module can prove this theorem automatically. *)
adam@315 319 (* begin hide *)
adam@315 321 (* end hide *)
adam@315 322 (** %\noindent \coqdockw{Restart}%#<tt>Restart</tt>#. *)
adamc@41 326 (* end thide *)
adamc@28 328 (** We can check out the induction principle at work here: *)
adamc@208 332 nat_ind : forall P : nat -> Prop,
adamc@208 333 P O -> (forall n : nat, P n -> P (S n)) -> forall n : nat, P n
adam@315 337 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.
adam@315 339 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@28 341 Theorem S_inj : forall n m : nat, S n = S m -> n = m.
adamc@41 342 (* begin thide *)
adamc@41 345 (* end thide *)
adamc@28 347 (** [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.
adam@315 349 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@29 353 We can define a type of lists of natural numbers. *)
adamc@29 355 Inductive nat_list : Set :=
adamc@29 356 | NNil : nat_list
adamc@29 357 | NCons : nat -> nat_list -> nat_list.
adamc@29 359 (** Recursive definitions are straightforward extensions of what we have seen before. *)
adamc@29 361 Fixpoint nlength (ls : nat_list) : nat :=
adamc@29 363 | NNil => O
adamc@29 364 | NCons _ ls' => S (nlength ls')
adamc@208 367 Fixpoint napp (ls1 ls2 : nat_list) : nat_list :=
adamc@29 369 | NNil => ls2
adamc@29 370 | NCons n ls1' => NCons n (napp ls1' ls2)
adamc@29 373 (** Inductive theorem proving can again be automated quite effectively. *)
adamc@29 375 Theorem nlength_napp : forall ls1 ls2 : nat_list, nlength (napp ls1 ls2)
adamc@29 376 = plus (nlength ls1) (nlength ls2).
adamc@41 377 (* begin thide *)
adamc@41 380 (* end thide *)
adamc@29 385 : forall P : nat_list -> Prop,
adamc@29 387 (forall (n : nat) (n0 : nat_list), P n0 -> P (NCons n n0)) ->
adamc@29 388 forall n : nat_list, P n
adam@292 393 In general, we can implement any %%#"#tree#"#%''% types as inductive types. For example, here are binary trees of naturals. *)
adamc@29 395 Inductive nat_btree : Set :=
adamc@29 396 | NLeaf : nat_btree
adamc@29 397 | NNode : nat_btree -> nat -> nat_btree -> nat_btree.
adamc@29 399 Fixpoint nsize (tr : nat_btree) : nat :=
adamc@35 401 | NLeaf => S O
adamc@29 402 | NNode tr1 _ tr2 => plus (nsize tr1) (nsize tr2)
adamc@208 405 Fixpoint nsplice (tr1 tr2 : nat_btree) : nat_btree :=
adamc@35 407 | NLeaf => NNode tr2 O NLeaf
adamc@29 408 | NNode tr1' n tr2' => NNode (nsplice tr1' tr2) n tr2'
adamc@29 411 Theorem plus_assoc : forall n1 n2 n3 : nat, plus (plus n1 n2) n3 = plus n1 (plus n2 n3).
adamc@41 412 (* begin thide *)
adamc@41 415 (* end thide *)
adamc@29 417 Theorem nsize_nsplice : forall tr1 tr2 : nat_btree, nsize (nsplice tr1 tr2)
adamc@29 418 = plus (nsize tr2) (nsize tr1).
adamc@41 419 (* begin thide *)
adam@315 420 (* begin hide *)
adamc@29 421 Hint Rewrite n_plus_O plus_assoc : cpdt.
adam@315 422 (* end hide *)
adam@315 423 (** [Hint] %\coqdockw{%#<tt>#Rewrite#</tt>#%}% [n_plus_O plus_assoc : cpdt.] *)
adamc@41 427 (* end thide *)
adam@315 429 (** It is convenient that these proofs go through so easily, but it is useful to check that the tree induction principle works as usual. *)
adamc@29 434 : forall P : nat_btree -> Prop,
adamc@29 436 (forall n : nat_btree,
adamc@29 437 P n -> forall (n0 : nat) (n1 : nat_btree), P n1 -> P (NNode n n0 n1)) ->
adamc@29 438 forall n : nat_btree, P n
adam@315 441 We have the usual two cases, one for each constructor of [nat_btree]. *)
adamc@30 444 (** * Parameterized Types *)
adamc@30 446 (** We can also define polymorphic inductive types, as with algebraic datatypes in Haskell and ML. *)
adamc@30 448 Inductive list (T : Set) : Set :=
adamc@30 449 | Nil : list T
adamc@30 450 | Cons : T -> list T -> list T.
adamc@30 452 Fixpoint length T (ls : list T) : nat :=
adamc@30 454 | Nil => O
adamc@30 455 | Cons _ ls' => S (length ls')
adamc@208 458 Fixpoint app T (ls1 ls2 : list T) : list T :=
adamc@30 460 | Nil => ls2
adamc@30 461 | Cons x ls1' => Cons x (app ls1' ls2)
adamc@30 464 Theorem length_app : forall T (ls1 ls2 : list T), length (app ls1 ls2)
adamc@30 465 = plus (length ls1) (length ls2).
adamc@41 466 (* begin thide *)
adamc@41 469 (* end thide *)
adamc@30 471 (** There is a useful shorthand for writing many definitions that share the same parameter, based on Coq's %\textit{%#<i>#section#</i>#%}% mechanism. The following block of code is equivalent to the above: *)
adamc@30 473 (* begin hide *)
adamc@30 475 (* end hide *)
adamc@30 478 Variable T : Set.
adamc@30 480 Inductive list : Set :=
adamc@30 481 | Nil : list
adamc@30 482 | Cons : T -> list -> list.
adamc@30 484 Fixpoint length (ls : list) : nat :=
adamc@30 486 | Nil => O
adamc@30 487 | Cons _ ls' => S (length ls')
adamc@208 490 Fixpoint app (ls1 ls2 : list) : list :=
adamc@30 492 | Nil => ls2
adamc@30 493 | Cons x ls1' => Cons x (app ls1' ls2)
adamc@30 496 Theorem length_app : forall ls1 ls2 : list, length (app ls1 ls2)
adamc@30 497 = plus (length ls1) (length ls2).
adamc@41 498 (* begin thide *)
adamc@41 501 (* end thide *)
adamc@35 504 (* begin hide *)
adamc@35 505 Implicit Arguments Nil [T].
adamc@35 506 (* end hide *)
adamc@210 508 (** 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@208 512 Inductive list (T : Set) : Set :=
adamc@202 513 Nil : list T | Cons : T -> list T -> list Tlist
adamc@202 517 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@30 522 : forall T : Set, list T -> nat
adamc@202 525 The parameter [T] is treated as a new argument to the induction principle, too. *)
adamc@30 530 : forall (T : Set) (P : list T -> Prop),
adamc@30 531 P (Nil T) ->
adamc@30 532 (forall (t : T) (l : list T), P l -> P (Cons t l)) ->
adamc@30 533 forall l : list T, P l
adamc@30 536 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 (** * Mutually Inductive Types *)
adamc@31 541 (** We can define inductive types that refer to each other: *)
adamc@31 543 Inductive even_list : Set :=
adamc@31 544 | ENil : even_list
adamc@31 545 | ECons : nat -> odd_list -> even_list
adamc@31 547 with odd_list : Set :=
adamc@31 548 | OCons : nat -> even_list -> odd_list.
adamc@31 550 Fixpoint elength (el : even_list) : nat :=
adamc@31 552 | ENil => O
adamc@31 553 | ECons _ ol => S (olength ol)
adamc@31 556 with olength (ol : odd_list) : nat :=
adamc@31 558 | OCons _ el => S (elength el)
adamc@208 561 Fixpoint eapp (el1 el2 : even_list) : even_list :=
adamc@31 563 | ENil => el2
adamc@31 564 | ECons n ol => ECons n (oapp ol el2)
adamc@208 567 with oapp (ol : odd_list) (el : even_list) : odd_list :=
adamc@31 569 | OCons n el' => OCons n (eapp el' el)
adamc@31 572 (** Everything is going roughly the same as in past examples, until we try to prove a theorem similar to those that came before. *)
adamc@31 574 Theorem elength_eapp : forall el1 el2 : even_list,
adamc@31 575 elength (eapp el1 el2) = plus (elength el1) (elength el2).
adamc@41 576 (* begin thide *)
adamc@31 579 (** One goal remains: [[
adamc@31 585 S (olength (oapp o el2)) = S (plus (olength o) (elength el2))
adamc@31 588 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@31 594 : forall P : even_list -> Prop,
adamc@31 596 (forall (n : nat) (o : odd_list), P (ECons n o)) ->
adamc@31 597 forall e : even_list, P e
adamc@31 601 We see that no inductive hypotheses are included anywhere in the type. To get them, we must ask for mutual principles as we need them, using the [Scheme] command. *)
adamc@31 603 Scheme even_list_mut := Induction for even_list Sort Prop
adamc@31 604 with odd_list_mut := Induction for odd_list Sort Prop.
adamc@31 609 : forall (P : even_list -> Prop) (P0 : odd_list -> Prop),
adamc@31 611 (forall (n : nat) (o : odd_list), P0 o -> P (ECons n o)) ->
adamc@31 612 (forall (n : nat) (e : even_list), P e -> P0 (OCons n e)) ->
adamc@31 613 forall e : even_list, P e
adamc@31 617 This is the principle we wanted in the first place. There is one more wrinkle left in using it: the [induction] tactic will not apply it for us automatically. It will be helpful to look at how to prove one of our past examples without using [induction], so that we can then generalize the technique to mutual inductive types. *)
adamc@31 619 Theorem n_plus_O' : forall n : nat, plus n O = n.
adamc@31 620 apply (nat_ind (fun n => plus n O = n)); crush.
adamc@31 623 (** 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@31 625 This technique generalizes to our mutual example: *)
adamc@31 627 Theorem elength_eapp : forall el1 el2 : even_list,
adamc@31 628 elength (eapp el1 el2) = plus (elength el1) (elength el2).
adamc@31 631 (fun el1 : even_list => forall el2 : even_list,
adamc@31 632 elength (eapp el1 el2) = plus (elength el1) (elength el2))
adamc@31 633 (fun ol : odd_list => forall el : even_list,
adamc@31 634 olength (oapp ol el) = plus (olength ol) (elength el))); crush.
adamc@41 636 (* end thide *)
adamc@31 638 (** 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@33 641 (** * Reflexive Types *)
adamc@33 643 (** A kind of inductive type called a %\textit{%#<i>#reflexive type#</i>#%}% is defined in terms of functions that have the type being defined as their range. One very useful class of examples is in modeling variable binders. For instance, here is a type for encoding the syntax of a subset of first-order logic: *)
adamc@33 645 Inductive formula : Set :=
adamc@33 646 | Eq : nat -> nat -> formula
adamc@33 647 | And : formula -> formula -> formula
adamc@33 648 | Forall : (nat -> formula) -> formula.
adam@292 650 (** Our kinds of formulas are equalities between naturals, conjunction, and universal quantification over natural numbers. We avoid needing to include a notion of %%#"#variables#"#%''% in our type, by using Coq functions to encode quantification. For instance, here is the encoding of [forall x : nat, x = x]: *)
adamc@33 652 Example forall_refl : formula := Forall (fun x => Eq x x).
adamc@33 654 (** We can write recursive functions over reflexive types quite naturally. Here is one translating our formulas into native Coq propositions. *)
adamc@33 658 | Eq n1 n2 => n1 = n2
adamc@33 660 | Forall f' => forall n : nat, formulaDenote (f' n)
adamc@33 663 (** We can also encode a trivial formula transformation that swaps the order of equality and conjunction operands. *)
adamc@33 665 Fixpoint swapper (f : formula) : formula :=
adamc@33 667 | Eq n1 n2 => Eq n2 n1
adamc@33 668 | And f1 f2 => And (swapper f2) (swapper f1)
adamc@33 669 | Forall f' => Forall (fun n => swapper (f' n))
adamc@33 672 (** It is helpful to prove that this transformation does not make true formulas false. *)
adamc@41 675 (* begin thide *)
adamc@41 678 (* end thide *)
adamc@33 680 (** We can take a look at the induction principle behind this proof. *)
adamc@33 685 : forall P : formula -> Prop,
adamc@33 686 (forall n n0 : nat, P (Eq n n0)) ->
adamc@33 687 (forall f0 : formula,
adamc@33 688 P f0 -> forall f1 : formula, P f1 -> P (And f0 f1)) ->
adamc@33 689 (forall f1 : nat -> formula,
adamc@33 690 (forall n : nat, P (f1 n)) -> P (Forall f1)) ->
adamc@33 691 forall f2 : formula, P f2
adamc@208 695 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@33 699 Up to this point, we have seen how to encode in Coq more and more of what is possible with algebraic datatypes in Haskell and ML. This may have given the inaccurate impression that inductive types are a strict extension of algebraic datatypes. In fact, Coq must rule out some types allowed by Haskell and ML, for reasons of soundness. Reflexive types provide our first good example of such a case.
adamc@33 701 Given our last example of an inductive type, many readers are probably eager to try encoding the syntax of lambda calculus. Indeed, the function-based representation technique that we just used, called %\textit{%#<i>#higher-order abstract syntax (HOAS)#</i>#%}%, is the representation of choice for lambda calculi in Twelf and in many applications implemented in Haskell and ML. Let us try to import that choice to Coq: *)
adamc@33 704 Inductive term : Set :=
adamc@33 705 | App : term -> term -> term
adamc@33 706 | Abs : (term -> term) -> term.
adamc@33 708 Error: Non strictly positive occurrence of "term" in "(term -> term) -> term"
adamc@33 712 We have run afoul of the %\textit{%#<i>#strict positivity requirement#</i>#%}% for inductive definitions, which says that the type being defined may not occur to the left of an arrow in the type of a constructor argument. It is important that the type of a constructor is viewed in terms of a series of arguments and a result, since obviously we need recursive occurrences to the lefts of the outermost arrows if we are to have recursive occurrences at all.
adamc@33 714 Why must Coq enforce this restriction? Imagine that our last definition had been accepted, allowing us to write this function:
adamc@33 717 Definition uhoh (t : term) : term :=
adamc@33 719 | Abs f => f t
adamc@33 720 | _ => t
adamc@33 725 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@33 727 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@33 729 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@34 732 (** * An Interlude on Proof Terms *)
adamc@34 734 (** As we have emphasized a few times already, Coq proofs are actually programs, written in the same language we have been using in our examples all along. We can get a first sense of what this means by taking a look at the definitions of some of the induction principles we have used. *)
adamc@208 739 fun P : unit -> Prop => unit_rect P
adamc@34 740 : forall P : unit -> Prop, P tt -> forall u : unit, P u
adamc@34 744 We see that this induction principle is defined in terms of a more general principle, [unit_rect]. *)
adamc@34 749 : forall P : unit -> Type, P tt -> forall u : unit, P u
adamc@34 753 [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@208 758 fun P : unit -> Set => unit_rect P
adamc@34 759 : forall P : unit -> Set, P tt -> forall u : unit, P u
adamc@34 763 This is identical to the definition for [unit_ind], except that we have substituted [Set] for [Prop]. For most inductive types [T], then, we get not just induction principles [T_ind], but also recursion principles [T_rec]. We can use [T_rec] to write recursive definitions without explicit [Fixpoint] recursion. For instance, the following two definitions are equivalent: *)
adamc@34 765 Definition always_O (u : unit) : nat :=
adamc@34 767 | tt => O
adamc@34 770 Definition always_O' (u : unit) : nat :=
adamc@34 771 unit_rec (fun _ : unit => nat) O u.
adamc@34 773 (** 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@208 778 fun (P : unit -> Type) (f : P tt) (u : unit) =>
adamc@208 779 match u as u0 return (P u0) with
adamc@208 780 | tt => f
adamc@34 782 : forall P : unit -> Type, P tt -> forall u : unit, P u
adamc@34 786 The only new feature we see is an [as] clause for a [match], which is used in concert with the [return] clause that we saw in the introduction. Since the type of the [match] is dependent on the value of the object being analyzed, we must give that object a name so that we can refer to it in the [return] clause.
adamc@34 788 To prove that [unit_rect] is nothing special, we can reimplement it manually. *)
adamc@34 790 Definition unit_rect' (P : unit -> Type) (f : P tt) (u : unit) :=
adamc@34 792 | tt => f
adamc@208 795 (** We rely on Coq's heuristics for inferring [match] annotations.
adamc@208 797 We can check the implementation of [nat_rect] as well: *)
adamc@208 802 fun (P : nat -> Type) (f : P O) (f0 : forall n : nat, P n -> P (S n)) =>
adamc@208 803 fix F (n : nat) : P n :=
adamc@208 804 match n as n0 return (P n0) with
adamc@208 805 | O => f
adamc@208 806 | S n0 => f0 n0 (F n0)
adamc@208 808 : forall P : nat -> Type,
adamc@208 809 P O -> (forall n : nat, P n -> P (S n)) -> forall n : nat, P n
adamc@208 812 Now we have an actual recursive definition. [fix] expressions are an anonymous form of [Fixpoint], just as [fun] expressions stand for anonymous non-recursive functions. Beyond that, the syntax of [fix] mirrors that of [Fixpoint]. We can understand the definition of [nat_rect] better by reimplementing [nat_ind] using sections. *)
adamc@208 815 (** First, we have the property of natural numbers that we aim to prove. *)
adamc@208 817 Variable P : nat -> Prop.
adamc@208 819 (** Then we require a proof of the [O] case. *)
adamc@208 821 Hypothesis O_case : P O.
adamc@208 823 (** Next is a proof of the [S] case, which may assume an inductive hypothesis. *)
adamc@208 825 Hypothesis S_case : forall n : nat, P n -> P (S n).
adamc@208 827 (** Finally, we define a recursive function to tie the pieces together. *)
adamc@208 829 Fixpoint nat_ind' (n : nat) : P n :=
adamc@208 831 | O => O_case
adamc@208 832 | S n' => S_case (nat_ind' n')
adamc@208 836 (** 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@208 840 We can also examine the definition of [even_list_mut], which we generated with [Scheme] for a mutually-recursive type. *)
adamc@208 845 fun (P : even_list -> Prop) (P0 : odd_list -> Prop)
adamc@208 846 (f : P ENil) (f0 : forall (n : nat) (o : odd_list), P0 o -> P (ECons n o))
adamc@208 847 (f1 : forall (n : nat) (e : even_list), P e -> P0 (OCons n e)) =>
adamc@208 848 fix F (e : even_list) : P e :=
adamc@208 849 match e as e0 return (P e0) with
adamc@208 850 | ENil => f
adamc@208 851 | ECons n o => f0 n o (F0 o)
adamc@208 853 with F0 (o : odd_list) : P0 o :=
adamc@208 854 match o as o0 return (P0 o0) with
adamc@208 855 | OCons n e => f1 n e (F e)
adamc@208 858 : forall (P : even_list -> Prop) (P0 : odd_list -> Prop),
adamc@208 860 (forall (n : nat) (o : odd_list), P0 o -> P (ECons n o)) ->
adamc@208 861 (forall (n : nat) (e : even_list), P e -> P0 (OCons n e)) ->
adamc@208 862 forall e : even_list, P e
adamc@208 866 We see a mutually-recursive [fix], with the different functions separated by [with] in the same way that they would be separated by [and] in ML. A final [for] clause identifies which of the mutually-recursive functions should be the final value of the [fix] expression. Using this definition as a template, we can reimplement [even_list_mut] directly. *)
adamc@208 869 (** First, we need the properties that we are proving. *)
adamc@208 871 Variable Peven : even_list -> Prop.
adamc@208 872 Variable Podd : odd_list -> Prop.
adamc@208 874 (** Next, we need proofs of the three cases. *)
adamc@208 876 Hypothesis ENil_case : Peven ENil.
adamc@208 877 Hypothesis ECons_case : forall (n : nat) (o : odd_list), Podd o -> Peven (ECons n o).
adamc@208 878 Hypothesis OCons_case : forall (n : nat) (e : even_list), Peven e -> Podd (OCons n e).
adamc@208 880 (** Finally, we define the recursive functions. *)
adamc@208 882 Fixpoint even_list_mut' (e : even_list) : Peven e :=
adamc@208 884 | ENil => ENil_case
adamc@208 885 | ECons n o => ECons_case n (odd_list_mut' o)
adamc@208 887 with odd_list_mut' (o : odd_list) : Podd o :=
adamc@208 889 | OCons n e => OCons_case n (even_list_mut' e)
adamc@34 893 (** 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@34 896 Variable P : formula -> Prop.
adamc@38 897 Hypothesis Eq_case : forall n1 n2 : nat, P (Eq n1 n2).
adamc@38 898 Hypothesis And_case : forall f1 f2 : formula,
adamc@34 899 P f1 -> P f2 -> P (And f1 f2).
adamc@38 900 Hypothesis Forall_case : forall f : nat -> formula,
adamc@34 901 (forall n : nat, P (f n)) -> P (Forall f).
adamc@34 903 Fixpoint formula_ind' (f : formula) : P f :=
adamc@34 905 | Eq n1 n2 => Eq_case n1 n2
adamc@34 906 | And f1 f2 => And_case (formula_ind' f1) (formula_ind' f2)
adamc@34 907 | Forall f' => Forall_case f' (fun n => formula_ind' (f' n))
adamc@35 912 (** * Nested Inductive Types *)
adamc@35 914 (** 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@35 916 Inductive nat_tree : Set :=
adamc@35 917 | NLeaf' : nat_tree
adamc@35 918 | NNode' : nat -> list nat_tree -> nat_tree.
adamc@35 920 (** This is an example of a %\textit{%#<i>#nested#</i>#%}% inductive type definition, because we use the type we are defining as an argument to a parametrized type family. Coq will not allow all such definitions; it effectively pretends that we are defining [nat_tree] mutually with a version of [list] specialized to [nat_tree], checking that the resulting expanded definition satisfies the usual rules. For instance, if we replaced [list] with a type family that used its parameter as a function argument, then the definition would be rejected as violating the positivity restriction.
adamc@35 922 Like we encountered for mutual inductive types, we find that the automatically-generated induction principle for [nat_tree] is too weak. *)
adamc@35 927 : forall P : nat_tree -> Prop,
adamc@35 929 (forall (n : nat) (l : list nat_tree), P (NNode' n l)) ->
adamc@35 930 forall n : nat_tree, P n
adamc@35 934 There is no command like [Scheme] that will implement an improved principle for us. In general, it takes creativity to figure out how to incorporate nested uses to different type families. Now that we know how to implement induction principles manually, we are in a position to apply just such creativity to this problem.
adamc@35 936 First, we will need an auxiliary definition, characterizing what it means for a property to hold of every element of a list. *)
adamc@35 939 Variable T : Set.
adamc@35 940 Variable P : T -> Prop.
adamc@35 942 Fixpoint All (ls : list T) : Prop :=
adamc@35 944 | Nil => True
adamc@35 945 | Cons h t => P h /\ All t
adamc@35 949 (** It will be useful to look at the definitions of [True] and [/\], since we will want to write manual proofs of them below. *)
adamc@208 953 Inductive True : Prop := I : True
adamc@35 957 That is, [True] is a proposition with exactly one proof, [I], which we may always supply trivially.
adamc@35 959 Finding the definition of [/\] takes a little more work. Coq supports user registration of arbitrary parsing rules, and it is such a rule that is letting us write [/\] instead of an application of some inductive type family. We can find the underlying inductive type with the [Locate] command. *)
adamc@208 964 "A /\ B" := and A B : type_scope
adamc@208 971 Inductive and (A : Prop) (B : Prop) : Prop := conj : A -> B -> A /\ B
adamc@208 972 For conj: Arguments A, B are implicit
adamc@208 973 For and: Argument scopes are [type_scope type_scope]
adamc@208 974 For conj: Argument scopes are [type_scope type_scope _ _]
adamc@35 978 In addition to the definition of [and] itself, we get information on implicit arguments and parsing rules for [and] and its constructor [conj]. We will ignore the parsing information for now. The implicit argument information tells us that we build a proof of a conjunction by calling the constructor [conj] on proofs of the conjuncts, with no need to include the types of those proofs as explicit arguments.
adamc@35 982 Now we create a section for our induction principle, following the same basic plan as in the last section of this chapter. *)
adamc@35 985 Variable P : nat_tree -> Prop.
adamc@38 987 Hypothesis NLeaf'_case : P NLeaf'.
adamc@38 988 Hypothesis NNode'_case : forall (n : nat) (ls : list nat_tree),
adamc@35 989 All P ls -> P (NNode' n ls).
adamc@35 991 (** A first attempt at writing the induction principle itself follows the intuition that nested inductive type definitions are expanded into mutual inductive definitions.
adamc@35 994 Fixpoint nat_tree_ind' (tr : nat_tree) : P tr :=
adamc@35 996 | NLeaf' => NLeaf'_case
adamc@35 997 | NNode' n ls => NNode'_case n ls (list_nat_tree_ind ls)
adamc@35 1000 with list_nat_tree_ind (ls : list nat_tree) : All P ls :=
adamc@35 1002 | Nil => I
adamc@35 1003 | Cons tr rest => conj (nat_tree_ind' tr) (list_nat_tree_ind rest)
adam@292 1008 Coq rejects this definition, saying %%#"#Recursive call to nat_tree_ind' has principal argument equal to "tr" instead of rest.#"#%''% There is no deep theoretical reason why this program should be rejected; Coq applies incomplete termination-checking heuristics, and it is necessary to learn a few of the most important rules. The term %%#"#nested inductive type#"#%''% hints at the solution to this particular problem. Just like true mutually-inductive types require mutually-recursive induction principles, nested types require nested recursion. *)
adamc@35 1010 Fixpoint nat_tree_ind' (tr : nat_tree) : P tr :=
adamc@35 1012 | NLeaf' => NLeaf'_case
adamc@35 1013 | NNode' n ls => NNode'_case n ls
adamc@35 1014 ((fix list_nat_tree_ind (ls : list nat_tree) : All P ls :=
adamc@35 1016 | Nil => I
adamc@35 1017 | Cons tr rest => conj (nat_tree_ind' tr) (list_nat_tree_ind rest)
adam@279 1021 (** 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 1025 (** 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 1028 Variables T T' : Set.
adamc@35 1029 Variable f : T -> T'.
adamc@35 1031 Fixpoint map (ls : list T) : list T' :=
adamc@35 1033 | Nil => Nil
adamc@35 1034 | Cons h t => Cons (f h) (map t)
adamc@35 1038 Fixpoint sum (ls : list nat) : nat :=
adamc@35 1040 | Nil => O
adamc@35 1041 | Cons h t => plus h (sum t)
adamc@35 1044 (** Now we can define a size function over our trees. *)
adamc@35 1046 Fixpoint ntsize (tr : nat_tree) : nat :=
adamc@35 1048 | NLeaf' => S O
adamc@35 1049 | NNode' _ trs => S (sum (map ntsize trs))
adamc@35 1052 (** 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@208 1054 Fixpoint ntsplice (tr1 tr2 : nat_tree) : nat_tree :=
adamc@35 1056 | NLeaf' => NNode' O (Cons tr2 Nil)
adamc@35 1057 | NNode' n Nil => NNode' n (Cons tr2 Nil)
adamc@35 1058 | NNode' n (Cons tr trs) => NNode' n (Cons (ntsplice tr tr2) trs)
adamc@35 1061 (** 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@41 1063 (* begin thide *)
adamc@35 1064 Lemma plus_S : forall n1 n2 : nat,
adamc@35 1065 plus n1 (S n2) = S (plus n1 n2).
adamc@41 1068 (* end thide *)
adamc@35 1070 (** Now we begin the proof of the theorem, adding the lemma [plus_S] as a hint. *)
adamc@35 1072 Theorem ntsize_ntsplice : forall tr1 tr2 : nat_tree, ntsize (ntsplice tr1 tr2)
adamc@35 1073 = plus (ntsize tr2) (ntsize tr1).
adamc@41 1074 (* begin thide *)
adamc@35 1075 Hint Rewrite plus_S : cpdt.
adamc@35 1077 (** We know that the standard induction principle is insufficient for the task, so we need to provide a [using] clause for the [induction] tactic to specify our alternate principle. *)
adamc@35 1079 induction tr1 using nat_tree_ind'; crush.
adamc@35 1081 (** One subgoal remains: [[
adamc@35 1083 ls : list nat_tree
adamc@35 1085 (fun tr1 : nat_tree =>
adamc@35 1086 forall tr2 : nat_tree,
adamc@35 1087 ntsize (ntsplice tr1 tr2) = plus (ntsize tr2) (ntsize tr1)) ls
adamc@35 1092 | Nil => NNode' n (Cons tr2 Nil)
adamc@35 1093 | Cons tr trs => NNode' n (Cons (ntsplice tr tr2) trs)
adamc@35 1094 end = S (plus (ntsize tr2) (sum (map ntsize ls)))
adamc@35 1098 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@36 1102 (** We can go further in automating the proof by exploiting the hint mechanism. *)
adamc@35 1105 Hint Extern 1 (ntsize (match ?LS with Nil => _ | Cons _ _ => _ end) = _) =>
adamc@35 1107 induction tr1 using nat_tree_ind'; crush.
adamc@41 1109 (* end thide *)
adamc@35 1111 (** 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@40 1113 The advantage of using the hint is not very clear here, because the original proof was so short. However, the hint has fundamentally improved the readability of our proof. Before, the proof referred to the local variable [ls], which has an automatically-generated name. To a human reading the proof script without stepping through it interactively, it was not clear where [ls] came from. The hint explains to the reader the process for choosing which variables to case analyze on, and the hint can continue working even if the rest of the proof structure changes significantly. *)
adamc@36 1118 (** It can be useful to understand how tactics like [discriminate] and [injection] work, so it is worth stepping through a manual proof of each kind. We will start with a proof fit for [discriminate]. *)
adamc@36 1120 Theorem true_neq_false : true <> false.
adamc@41 1122 (* begin thide *)
adam@292 1123 (** We begin with the tactic [red], which is short for %%#"#one step of reduction,#"#%''% to unfold the definition of logical negation. *)
adamc@36 1128 true = false -> False
adamc@36 1132 The negation is replaced with an implication of falsehood. We use the tactic [intro H] to change the assumption of the implication into a hypothesis named [H]. *)
adamc@36 1136 H : true = false
adamc@36 1142 This is the point in the proof where we apply some creativity. We define a function whose utility will become clear soon. *)
adamc@36 1144 Definition f (b : bool) := if b then True else False.
adamc@36 1146 (** It is worth recalling the difference between the lowercase and uppercase versions of truth and falsehood: [True] and [False] are logical propositions, while [true] and [false] are boolean values that we can case-analyze. We have defined [f] such that our conclusion of [False] is computationally equivalent to [f false]. Thus, the [change] tactic will let us change the conclusion to [f false]. *)
adamc@36 1150 H : true = false
adamc@202 1156 Now the righthand side of [H]'s equality appears in the conclusion, so we can rewrite, using the notation [<-] to request to replace the righthand side the equality with the lefthand side. *)
adamc@36 1160 H : true = false
adamc@36 1166 We are almost done. Just how close we are to done is revealed by computational simplification. *)
adamc@36 1170 H : true = false
adamc@41 1179 (* end thide *)
adamc@36 1181 (** I have no trivial automated version of this proof to suggest, beyond using [discriminate] or [congruence] in the first place.
adamc@36 1185 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 1187 Theorem S_inj' : forall n m : nat, S n = S m -> n = m.
adamc@41 1188 (* begin thide *)
adamc@36 1189 intros n m H.
adamc@36 1190 change (pred (S n) = pred (S m)).
adamc@41 1194 (* end thide *)
adamc@37 1197 (** * Exercises *)
adam@292 1201 %\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>#
adam@292 1205 %\item%#<li># Reimplement the second example language of Chapter 2 to use mutually-inductive types instead of dependent types. That is, define two separate (non-dependent) inductive types [nat_exp] and [bool_exp] for expressions of the two different types, rather than a single indexed type. To keep things simple, you may consider only the binary operators that take naturals as operands. Add natural number variables to the language, as in the last exercise, and add an %%#"#if#"#%''% expression form taking as arguments one boolean expression and two natural number expressions. Define semantics and constant-folding functions for this new language. Your constant folding should simplify not just binary operations (returning naturals or booleans) with known arguments, but also %%#"#if#"#%''% expressions with known values for their test expressions but possibly undetermined %%#"#then#"#%''% and %%#"#else#"#%''% cases. Prove that constant-folding a natural number expression preserves its meaning.#</li>#