annotate src/GeneralRec.v @ 465:4320c1a967c2

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author Adam Chlipala <adam@chlipala.net>
date Wed, 29 Aug 2012 18:26:26 -0400
parents e3fa35c750ac
children b36876d4611e
rev   line source
adam@380 1 (* Copyright (c) 2006, 2011-2012, Adam Chlipala
adam@350 2 *
adam@350 3 * This work is licensed under a
adam@350 4 * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
adam@350 5 * Unported License.
adam@350 6 * The license text is available at:
adam@350 7 * http://creativecommons.org/licenses/by-nc-nd/3.0/
adam@350 8 *)
adam@350 9
adam@350 10 (* begin hide *)
adam@351 11 Require Import Arith List.
adam@350 12
adam@351 13 Require Import CpdtTactics Coinductive.
adam@350 14
adam@350 15 Set Implicit Arguments.
adam@350 16 (* end hide *)
adam@350 17
adam@350 18
adam@350 19 (** %\chapter{General Recursion}% *)
adam@350 20
adam@353 21 (** Termination of all programs is a crucial property of Gallina. Non-terminating programs introduce logical inconsistency, where any theorem can be proved with an infinite loop. Coq uses a small set of conservative, syntactic criteria to check termination of all recursive definitions. These criteria are insufficient to support the natural encodings of a variety of important programming idioms. Further, since Coq makes it so convenient to encode mathematics computationally, with functional programs, we may find ourselves wanting to employ more complicated recursion in mathematical definitions.
adam@351 22
adam@424 23 What exactly are the conservative criteria that we run up against? For _recursive_ definitions, recursive calls are only allowed on _syntactic subterms_ of the original primary argument, a restriction known as%\index{primitive recursion}% _primitive recursion_. In fact, Coq's handling of reflexive inductive types (those defined in terms of functions returning the same type) gives a bit more flexibility than in traditional primitive recursion, but the term is still applied commonly. In Chapter 5, we saw how _co-recursive_ definitions are checked against a syntactic guardedness condition that guarantees productivity.
adam@351 24
adam@353 25 Many natural recursion patterns satisfy neither condition. For instance, there is our simple running example in this chapter, merge sort. We will study three different approaches to more flexible recursion, and the latter two of the approaches will even support definitions that may fail to terminate on certain inputs, without any up-front characterization of which inputs those may be.
adam@351 26
adam@404 27 Before proceeding, it is important to note that the problem here is not as fundamental as it may appear. The final example of Chapter 5 demonstrated what is called a%\index{deep embedding}% _deep embedding_ of the syntax and semantics of a programming language. That is, we gave a mathematical definition of a language of programs and their meanings. This language clearly admitted non-termination, and we could think of writing all our sophisticated recursive functions with such explicit syntax types. However, in doing so, we forfeit our chance to take advantage of Coq's very good built-in support for reasoning about Gallina programs. We would rather use a%\index{shallow embedding}% _shallow embedding_, where we model informal constructs by encoding them as normal Gallina programs. Each of the three techniques of this chapter follows that style. *)
adam@351 28
adam@351 29
adam@351 30 (** * Well-Founded Recursion *)
adam@351 31
adam@404 32 (** The essence of terminating recursion is that there are no infinite chains of nested recursive calls. This intuition is commonly mapped to the mathematical idea of a%\index{well-founded relation}% _well-founded relation_, and the associated standard technique in Coq is%\index{well-founded recursion}% _well-founded recursion_. The syntactic-subterm relation that Coq applies by default is well-founded, but many cases demand alternate well-founded relations. To demonstrate, let us see where we get stuck on attempting a standard merge sort implementation. *)
adam@351 33
adam@351 34 Section mergeSort.
adam@351 35 Variable A : Type.
adam@351 36 Variable le : A -> A -> bool.
adam@424 37 (** We have a set equipped with some "less-than-or-equal-to" test. *)
adam@351 38
adam@351 39 (** A standard function inserts an element into a sorted list, preserving sortedness. *)
adam@351 40
adam@351 41 Fixpoint insert (x : A) (ls : list A) : list A :=
adam@351 42 match ls with
adam@351 43 | nil => x :: nil
adam@351 44 | h :: ls' =>
adam@351 45 if le x h
adam@351 46 then x :: ls
adam@351 47 else h :: insert x ls'
adam@351 48 end.
adam@351 49
adam@351 50 (** We will also need a function to merge two sorted lists. (We use a less efficient implementation than usual, because the more efficient implementation already forces us to think about well-founded recursion, while here we are only interested in setting up the example of merge sort.) *)
adam@351 51
adam@351 52 Fixpoint merge (ls1 ls2 : list A) : list A :=
adam@351 53 match ls1 with
adam@351 54 | nil => ls2
adam@351 55 | h :: ls' => insert h (merge ls' ls2)
adam@351 56 end.
adam@351 57
adam@351 58 (** The last helper function for classic merge sort is the one that follows, to partition a list arbitrarily into two pieces of approximately equal length. *)
adam@351 59
adam@351 60 Fixpoint partition (ls : list A) : list A * list A :=
adam@351 61 match ls with
adam@351 62 | nil => (nil, nil)
adam@351 63 | h :: nil => (h :: nil, nil)
adam@351 64 | h1 :: h2 :: ls' =>
adam@351 65 let (ls1, ls2) := partition ls' in
adam@351 66 (h1 :: ls1, h2 :: ls2)
adam@351 67 end.
adam@351 68
adam@424 69 (** Now, let us try to write the final sorting function, using a natural number "[<=]" test [leb] from the standard library.
adam@351 70 [[
adam@351 71 Fixpoint mergeSort (ls : list A) : list A :=
adam@453 72 if leb (length ls) 1
adam@351 73 then ls
adam@351 74 else let lss := partition ls in
adam@351 75 merge (mergeSort (fst lss)) (mergeSort (snd lss)).
adam@351 76 ]]
adam@351 77
adam@351 78 <<
adam@351 79 Recursive call to mergeSort has principal argument equal to
adam@351 80 "fst (partition ls)" instead of a subterm of "ls".
adam@351 81 >>
adam@351 82
adam@351 83 The definition is rejected for not following the simple primitive recursion criterion. In particular, it is not apparent that recursive calls to [mergeSort] are syntactic subterms of the original argument [ls]; indeed, they are not, yet we know this is a well-founded recursive definition.
adam@351 84
adam@351 85 To produce an acceptable definition, we need to choose a well-founded relation and prove that [mergeSort] respects it. A good starting point is an examination of how well-foundedness is formalized in the Coq standard library. *)
adam@351 86
adam@351 87 Print well_founded.
adam@351 88 (** %\vspace{-.15in}% [[
adam@351 89 well_founded =
adam@351 90 fun (A : Type) (R : A -> A -> Prop) => forall a : A, Acc R a
adam@351 91 ]]
adam@351 92
adam@404 93 The bulk of the definitional work devolves to the%\index{accessibility relation}\index{Gallina terms!Acc}% _accessibility_ relation [Acc], whose definition we may also examine. *)
adam@351 94
adam@424 95 (* begin hide *)
adam@437 96 (* begin thide *)
adam@424 97 Definition Acc_intro' := Acc_intro.
adam@437 98 (* end thide *)
adam@424 99 (* end hide *)
adam@424 100
adam@351 101 Print Acc.
adam@351 102 (** %\vspace{-.15in}% [[
adam@351 103 Inductive Acc (A : Type) (R : A -> A -> Prop) (x : A) : Prop :=
adam@351 104 Acc_intro : (forall y : A, R y x -> Acc R y) -> Acc R x
adam@351 105 ]]
adam@351 106
adam@424 107 In prose, an element [x] is accessible for a relation [R] if every element "less than" [x] according to [R] is also accessible. Since [Acc] is defined inductively, we know that any accessibility proof involves a finite chain of invocations, in a certain sense that we can make formal. Building on Chapter 5's examples, let us define a co-inductive relation that is closer to the usual informal notion of "absence of infinite decreasing chains." *)
adam@351 108
adam@351 109 CoInductive isChain A (R : A -> A -> Prop) : stream A -> Prop :=
adam@351 110 | ChainCons : forall x y s, isChain R (Cons y s)
adam@351 111 -> R y x
adam@351 112 -> isChain R (Cons x (Cons y s)).
adam@351 113
adam@351 114 (** We can now prove that any accessible element cannot be the beginning of any infinite decreasing chain. *)
adam@351 115
adam@351 116 (* begin thide *)
adam@351 117 Lemma noChains' : forall A (R : A -> A -> Prop) x, Acc R x
adam@351 118 -> forall s, ~isChain R (Cons x s).
adam@351 119 induction 1; crush;
adam@351 120 match goal with
adam@351 121 | [ H : isChain _ _ |- _ ] => inversion H; eauto
adam@351 122 end.
adam@351 123 Qed.
adam@351 124
adam@351 125 (** From here, the absence of infinite decreasing chains in well-founded sets is immediate. *)
adam@351 126
adam@351 127 Theorem noChains : forall A (R : A -> A -> Prop), well_founded R
adam@351 128 -> forall s, ~isChain R s.
adam@351 129 destruct s; apply noChains'; auto.
adam@351 130 Qed.
adam@351 131 (* end thide *)
adam@351 132
adam@351 133 (** Absence of infinite decreasing chains implies absence of infinitely nested recursive calls, for any recursive definition that respects the well-founded relation. The [Fix] combinator from the standard library formalizes that intuition: *)
adam@351 134
adam@351 135 Check Fix.
adam@351 136 (** %\vspace{-.15in}%[[
adam@351 137 Fix
adam@351 138 : forall (A : Type) (R : A -> A -> Prop),
adam@351 139 well_founded R ->
adam@351 140 forall P : A -> Type,
adam@351 141 (forall x : A, (forall y : A, R y x -> P y) -> P x) ->
adam@351 142 forall x : A, P x
adam@351 143 ]]
adam@351 144
adam@351 145 A call to %\index{Gallina terms!Fix}%[Fix] must present a relation [R] and a proof of its well-foundedness. The next argument, [P], is the possibly dependent range type of the function we build; the domain [A] of [R] is the function's domain. The following argument has this type:
adam@351 146 [[
adam@351 147 forall x : A, (forall y : A, R y x -> P y) -> P x
adam@351 148 ]]
adam@351 149
adam@424 150 This is an encoding of the function body. The input [x] stands for the function argument, and the next input stands for the function we are defining. Recursive calls are encoded as calls to the second argument, whose type tells us it expects a value [y] and a proof that [y] is "less than" [x], according to [R]. In this way, we enforce the well-foundedness restriction on recursive calls.
adam@351 151
adam@353 152 The rest of [Fix]'s type tells us that it returns a function of exactly the type we expect, so we are now ready to use it to implement [mergeSort]. Careful readers may have noticed that [Fix] has a dependent type of the sort we met in the previous chapter.
adam@351 153
adam@351 154 Before writing [mergeSort], we need to settle on a well-founded relation. The right one for this example is based on lengths of lists. *)
adam@351 155
adam@351 156 Definition lengthOrder (ls1 ls2 : list A) :=
adam@351 157 length ls1 < length ls2.
adam@351 158
adam@353 159 (** We must prove that the relation is truly well-founded. To save some space in the rest of this chapter, we skip right to nice, automated proof scripts, though we postpone introducing the principles behind such scripts to Part III of the book. Curious readers may still replace semicolons with periods and newlines to step through these scripts interactively. *)
adam@351 160
adam@351 161 Hint Constructors Acc.
adam@351 162
adam@351 163 Lemma lengthOrder_wf' : forall len, forall ls, length ls <= len -> Acc lengthOrder ls.
adam@351 164 unfold lengthOrder; induction len; crush.
adam@351 165 Defined.
adam@351 166
adam@351 167 Theorem lengthOrder_wf : well_founded lengthOrder.
adam@351 168 red; intro; eapply lengthOrder_wf'; eauto.
adam@351 169 Defined.
adam@351 170
adam@398 171 (** Notice that we end these proofs with %\index{Vernacular commands!Defined}%[Defined], not [Qed]. Recall that [Defined] marks the theorems as %\emph{transparent}%, so that the details of their proofs may be used during program execution. Why could such details possibly matter for computation? It turns out that [Fix] satisfies the primitive recursion restriction by declaring itself as _recursive in the structure of [Acc] proofs_. This is possible because [Acc] proofs follow a predictable inductive structure. We must do work, as in the last theorem's proof, to establish that all elements of a type belong to [Acc], but the automatic unwinding of those proofs during recursion is straightforward. If we ended the proof with [Qed], the proof details would be hidden from computation, in which case the unwinding process would get stuck.
adam@351 172
adam@351 173 To justify our two recursive [mergeSort] calls, we will also need to prove that [partition] respects the [lengthOrder] relation. These proofs, too, must be kept transparent, to avoid stuckness of [Fix] evaluation. *)
adam@351 174
adam@351 175 Lemma partition_wf : forall len ls, 2 <= length ls <= len
adam@351 176 -> let (ls1, ls2) := partition ls in
adam@351 177 lengthOrder ls1 ls /\ lengthOrder ls2 ls.
adam@351 178 unfold lengthOrder; induction len; crush; do 2 (destruct ls; crush);
adam@351 179 destruct (le_lt_dec 2 (length ls));
adam@351 180 repeat (match goal with
adam@351 181 | [ _ : length ?E < 2 |- _ ] => destruct E
adam@351 182 | [ _ : S (length ?E) < 2 |- _ ] => destruct E
adam@351 183 | [ IH : _ |- context[partition ?L] ] =>
adam@351 184 specialize (IH L); destruct (partition L); destruct IH
adam@351 185 end; crush).
adam@351 186 Defined.
adam@351 187
adam@351 188 Ltac partition := intros ls ?; intros; generalize (@partition_wf (length ls) ls);
adam@351 189 destruct (partition ls); destruct 1; crush.
adam@351 190
adam@351 191 Lemma partition_wf1 : forall ls, 2 <= length ls
adam@351 192 -> lengthOrder (fst (partition ls)) ls.
adam@351 193 partition.
adam@351 194 Defined.
adam@351 195
adam@351 196 Lemma partition_wf2 : forall ls, 2 <= length ls
adam@351 197 -> lengthOrder (snd (partition ls)) ls.
adam@351 198 partition.
adam@351 199 Defined.
adam@351 200
adam@351 201 Hint Resolve partition_wf1 partition_wf2.
adam@351 202
adam@453 203 (** To write the function definition itself, we use the %\index{tactics!refine}%[refine] tactic as a convenient way to write a program that needs to manipulate proofs, without writing out those proofs manually. We also use a replacement [le_lt_dec] for [leb] that has a more interesting dependent type. (Note that we would not be able to complete the definition without this change, since [refine] will generate subgoals for the [if] branches based only on the _type_ of the test expression, not its _value_.) *)
adam@351 204
adam@351 205 Definition mergeSort : list A -> list A.
adam@351 206 (* begin thide *)
adam@351 207 refine (Fix lengthOrder_wf (fun _ => list A)
adam@351 208 (fun (ls : list A)
adam@351 209 (mergeSort : forall ls' : list A, lengthOrder ls' ls -> list A) =>
adam@351 210 if le_lt_dec 2 (length ls)
adam@351 211 then let lss := partition ls in
adam@351 212 merge (mergeSort (fst lss) _) (mergeSort (snd lss) _)
adam@351 213 else ls)); subst lss; eauto.
adam@351 214 Defined.
adam@351 215 (* end thide *)
adam@351 216 End mergeSort.
adam@351 217
adam@351 218 (** The important thing is that it is now easy to evaluate calls to [mergeSort]. *)
adam@351 219
adam@351 220 Eval compute in mergeSort leb (1 :: 2 :: 36 :: 8 :: 19 :: nil).
adam@351 221 (** [= 1 :: 2 :: 8 :: 19 :: 36 :: nil] *)
adam@351 222
adam@441 223 (** %\smallskip{}%Since the subject of this chapter is merely how to define functions with unusual recursion structure, we will not prove any further correctness theorems about [mergeSort]. Instead, we stop at proving that [mergeSort] has the expected computational behavior, for all inputs, not merely the one we just tested. *)
adam@351 224
adam@351 225 (* begin thide *)
adam@351 226 Theorem mergeSort_eq : forall A (le : A -> A -> bool) ls,
adam@351 227 mergeSort le ls = if le_lt_dec 2 (length ls)
adam@351 228 then let lss := partition ls in
adam@351 229 merge le (mergeSort le (fst lss)) (mergeSort le (snd lss))
adam@351 230 else ls.
adam@351 231 intros; apply (Fix_eq (@lengthOrder_wf A) (fun _ => list A)); intros.
adam@351 232
adam@424 233 (** The library theorem [Fix_eq] imposes one more strange subgoal upon us. We must prove that the function body is unable to distinguish between "self" arguments that map equal inputs to equal outputs. One might think this should be true of any Gallina code, but in fact this general%\index{extensionality}% _function extensionality_ property is neither provable nor disprovable within Coq. The type of [Fix_eq] makes clear what we must show manually: *)
adam@351 234
adam@351 235 Check Fix_eq.
adam@351 236 (** %\vspace{-.15in}%[[
adam@351 237 Fix_eq
adam@351 238 : forall (A : Type) (R : A -> A -> Prop) (Rwf : well_founded R)
adam@351 239 (P : A -> Type)
adam@351 240 (F : forall x : A, (forall y : A, R y x -> P y) -> P x),
adam@351 241 (forall (x : A) (f g : forall y : A, R y x -> P y),
adam@351 242 (forall (y : A) (p : R y x), f y p = g y p) -> F x f = F x g) ->
adam@351 243 forall x : A,
adam@351 244 Fix Rwf P F x = F x (fun (y : A) (_ : R y x) => Fix Rwf P F y)
adam@351 245 ]]
adam@351 246
adam@465 247 Most such obligations are dischargeable with straightforward proof automation, and this example is no exception. *)
adam@351 248
adam@351 249 match goal with
adam@351 250 | [ |- context[match ?E with left _ => _ | right _ => _ end] ] => destruct E
adam@351 251 end; simpl; f_equal; auto.
adam@351 252 Qed.
adam@351 253 (* end thide *)
adam@351 254
adam@351 255 (** As a final test of our definition's suitability, we can extract to OCaml. *)
adam@351 256
adam@351 257 Extraction mergeSort.
adam@351 258
adam@351 259 (** <<
adam@351 260 let rec mergeSort le x =
adam@351 261 match le_lt_dec (S (S O)) (length x) with
adam@351 262 | Left ->
adam@351 263 let lss = partition x in
adam@351 264 merge le (mergeSort le (fst lss)) (mergeSort le (snd lss))
adam@351 265 | Right -> x
adam@351 266 >>
adam@351 267
adam@353 268 We see almost precisely the same definition we would have written manually in OCaml! It might be a good exercise for the reader to use the commands we saw in the previous chapter to clean up some remaining differences from idiomatic OCaml.
adam@351 269
adam@351 270 One more piece of the full picture is missing. To go on and prove correctness of [mergeSort], we would need more than a way of unfolding its definition. We also need an appropriate induction principle matched to the well-founded relation. Such a principle is available in the standard library, though we will say no more about its details here. *)
adam@351 271
adam@351 272 Check well_founded_induction.
adam@351 273 (** %\vspace{-.15in}%[[
adam@351 274 well_founded_induction
adam@351 275 : forall (A : Type) (R : A -> A -> Prop),
adam@351 276 well_founded R ->
adam@351 277 forall P : A -> Set,
adam@351 278 (forall x : A, (forall y : A, R y x -> P y) -> P x) ->
adam@351 279 forall a : A, P a
adam@351 280 ]]
adam@351 281
adam@351 282 Some more recent Coq features provide more convenient syntax for defining recursive functions. Interested readers can consult the Coq manual about the commands %\index{Function}%[Function] and %\index{Program Fixpoint}%[Program Fixpoint]. *)
adam@352 283
adam@352 284
adam@354 285 (** * A Non-Termination Monad Inspired by Domain Theory *)
adam@352 286
adam@424 287 (** The key insights of %\index{domain theory}%domain theory%~\cite{WinskelDomains}% inspire the next approach to modeling non-termination. Domain theory is based on _information orders_ that relate values representing computation results, according to how much information these values convey. For instance, a simple domain might include values "the program does not terminate" and "the program terminates with the answer 5." The former is considered to be an _approximation_ of the latter, while the latter is _not_ an approximation of "the program terminates with the answer 6." The details of domain theory will not be important in what follows; we merely borrow the notion of an approximation ordering on computation results.
adam@355 288
adam@355 289 Consider this definition of a type of computations. *)
adam@355 290
adam@352 291 Section computation.
adam@352 292 Variable A : Type.
adam@355 293 (** The type [A] describes the result a computation will yield, if it terminates.
adam@355 294
adam@355 295 We give a rich dependent type to computations themselves: *)
adam@352 296
adam@352 297 Definition computation :=
adam@352 298 {f : nat -> option A
adam@352 299 | forall (n : nat) (v : A),
adam@352 300 f n = Some v
adam@352 301 -> forall (n' : nat), n' >= n
adam@352 302 -> f n' = Some v}.
adam@352 303
adam@398 304 (** A computation is fundamentally a function [f] from an _approximation level_ [n] to an optional result. Intuitively, higher [n] values enable termination in more cases than lower values. A call to [f] may return [None] to indicate that [n] was not high enough to run the computation to completion; higher [n] values may yield [Some]. Further, the proof obligation within the sigma type asserts that [f] is _monotone_ in an appropriate sense: when some [n] is sufficient to produce termination, so are all higher [n] values, and they all yield the same program result [v].
adam@355 305
adam@355 306 It is easy to define a relation characterizing when a computation runs to a particular result at a particular approximation level. *)
adam@355 307
adam@352 308 Definition runTo (m : computation) (n : nat) (v : A) :=
adam@352 309 proj1_sig m n = Some v.
adam@352 310
adam@355 311 (** On top of [runTo], we also define [run], which is the most abstract notion of when a computation runs to a value. *)
adam@355 312
adam@352 313 Definition run (m : computation) (v : A) :=
adam@352 314 exists n, runTo m n v.
adam@352 315 End computation.
adam@352 316
adam@355 317 (** The book source code contains at this point some tactics, lemma proofs, and hint commands, to be used in proving facts about computations. Since their details are orthogonal to the message of this chapter, I have omitted them in the rendered version. *)
adam@355 318 (* begin hide *)
adam@355 319
adam@352 320 Hint Unfold runTo.
adam@352 321
adam@352 322 Ltac run' := unfold run, runTo in *; try red; crush;
adam@352 323 repeat (match goal with
adam@352 324 | [ _ : proj1_sig ?E _ = _ |- _ ] =>
adam@352 325 match goal with
adam@352 326 | [ x : _ |- _ ] =>
adam@352 327 match x with
adam@352 328 | E => destruct E
adam@352 329 end
adam@352 330 end
adam@352 331 | [ |- context[match ?M with exist _ _ => _ end] ] => let Heq := fresh "Heq" in
adam@352 332 case_eq M; intros ? ? Heq; try rewrite Heq in *; try subst
adam@352 333 | [ _ : context[match ?M with exist _ _ => _ end] |- _ ] => let Heq := fresh "Heq" in
adam@352 334 case_eq M; intros ? ? Heq; try rewrite Heq in *; subst
adam@352 335 | [ H : forall n v, ?E n = Some v -> _,
adam@352 336 _ : context[match ?E ?N with Some _ => _ | None => _ end] |- _ ] =>
adam@426 337 specialize (H N); destruct (E N); try rewrite (H _ (eq_refl _)) by auto; try discriminate
adam@352 338 | [ H : forall n v, ?E n = Some v -> _, H' : ?E _ = Some _ |- _ ] => rewrite (H _ _ H') by auto
adam@352 339 end; simpl in *); eauto 7.
adam@352 340
adam@352 341 Ltac run := run'; repeat (match goal with
adam@352 342 | [ H : forall n v, ?E n = Some v -> _
adam@352 343 |- context[match ?E ?N with Some _ => _ | None => _ end] ] =>
adam@426 344 specialize (H N); destruct (E N); try rewrite (H _ (eq_refl _)) by auto; try discriminate
adam@352 345 end; run').
adam@352 346
adam@352 347 Lemma ex_irrelevant : forall P : Prop, P -> exists n : nat, P.
adam@352 348 exists 0; auto.
adam@352 349 Qed.
adam@352 350
adam@352 351 Hint Resolve ex_irrelevant.
adam@352 352
adam@352 353 Require Import Max.
adam@352 354
adam@380 355 Theorem max_spec_le : forall n m, n <= m /\ max n m = m \/ m <= n /\ max n m = n.
adam@380 356 induction n; destruct m; simpl; intuition;
adam@380 357 specialize (IHn m); intuition.
adam@380 358 Qed.
adam@380 359
adam@352 360 Ltac max := intros n m; generalize (max_spec_le n m); crush.
adam@352 361
adam@352 362 Lemma max_1 : forall n m, max n m >= n.
adam@352 363 max.
adam@352 364 Qed.
adam@352 365
adam@352 366 Lemma max_2 : forall n m, max n m >= m.
adam@352 367 max.
adam@352 368 Qed.
adam@352 369
adam@352 370 Hint Resolve max_1 max_2.
adam@352 371
adam@352 372 Lemma ge_refl : forall n, n >= n.
adam@352 373 crush.
adam@352 374 Qed.
adam@352 375
adam@352 376 Hint Resolve ge_refl.
adam@352 377
adam@352 378 Hint Extern 1 => match goal with
adam@352 379 | [ H : _ = exist _ _ _ |- _ ] => rewrite H
adam@352 380 end.
adam@355 381 (* end hide *)
adam@355 382 (** remove printing exists *)
adam@355 383
adam@357 384 (** Now, as a simple first example of a computation, we can define [Bottom], which corresponds to an infinite loop. For any approximation level, it fails to terminate (returns [None]). Note the use of [abstract] to create a new opaque lemma for the proof found by the [run] tactic. In contrast to the previous section, opaque proofs are fine here, since the proof components of computations do not influence evaluation behavior. *)
adam@352 385
adam@352 386 Section Bottom.
adam@352 387 Variable A : Type.
adam@352 388
adam@352 389 Definition Bottom : computation A.
adam@352 390 exists (fun _ : nat => @None A); abstract run.
adam@352 391 Defined.
adam@352 392
adam@352 393 Theorem run_Bottom : forall v, ~run Bottom v.
adam@352 394 run.
adam@352 395 Qed.
adam@352 396 End Bottom.
adam@352 397
adam@355 398 (** A slightly more complicated example is [Return], which gives the same terminating answer at every approximation level. *)
adam@355 399
adam@352 400 Section Return.
adam@352 401 Variable A : Type.
adam@352 402 Variable v : A.
adam@352 403
adam@352 404 Definition Return : computation A.
adam@352 405 intros; exists (fun _ : nat => Some v); abstract run.
adam@352 406 Defined.
adam@352 407
adam@352 408 Theorem run_Return : run Return v.
adam@352 409 run.
adam@352 410 Qed.
adam@352 411 End Return.
adam@352 412
adam@356 413 (** The name [Return] was meant to be suggestive of the standard operations of %\index{monad}%monads%~\cite{Monads}%. The other standard operation is [Bind], which lets us run one computation and, if it terminates, pass its result off to another computation. *)
adam@352 414
adam@352 415 Section Bind.
adam@352 416 Variables A B : Type.
adam@352 417 Variable m1 : computation A.
adam@352 418 Variable m2 : A -> computation B.
adam@352 419
adam@352 420 Definition Bind : computation B.
adam@352 421 exists (fun n =>
adam@357 422 let (f1, _) := m1 in
adam@352 423 match f1 n with
adam@352 424 | None => None
adam@352 425 | Some v =>
adam@357 426 let (f2, _) := m2 v in
adam@352 427 f2 n
adam@352 428 end); abstract run.
adam@352 429 Defined.
adam@352 430
adam@352 431 Theorem run_Bind : forall (v1 : A) (v2 : B),
adam@352 432 run m1 v1
adam@352 433 -> run (m2 v1) v2
adam@352 434 -> run Bind v2.
adam@352 435 run; match goal with
adam@352 436 | [ x : nat, y : nat |- _ ] => exists (max x y)
adam@352 437 end; run.
adam@352 438 Qed.
adam@352 439 End Bind.
adam@352 440
adam@355 441 (** A simple notation lets us write [Bind] calls the way they appear in Haskell. *)
adam@352 442
adam@352 443 Notation "x <- m1 ; m2" :=
adam@352 444 (Bind m1 (fun x => m2)) (right associativity, at level 70).
adam@352 445
adam@424 446 (** We can verify that we have indeed defined a monad, by proving the standard monad laws. Part of the exercise is choosing an appropriate notion of equality between computations. We use "equality at all approximation levels." *)
adam@355 447
adam@352 448 Definition meq A (m1 m2 : computation A) := forall n, proj1_sig m1 n = proj1_sig m2 n.
adam@352 449
adam@352 450 Theorem left_identity : forall A B (a : A) (f : A -> computation B),
adam@352 451 meq (Bind (Return a) f) (f a).
adam@352 452 run.
adam@352 453 Qed.
adam@352 454
adam@352 455 Theorem right_identity : forall A (m : computation A),
adam@352 456 meq (Bind m (@Return _)) m.
adam@352 457 run.
adam@352 458 Qed.
adam@352 459
adam@357 460 Theorem associativity : forall A B C (m : computation A)
adam@357 461 (f : A -> computation B) (g : B -> computation C),
adam@352 462 meq (Bind (Bind m f) g) (Bind m (fun x => Bind (f x) g)).
adam@352 463 run.
adam@352 464 Qed.
adam@352 465
adam@398 466 (** Now we come to the piece most directly inspired by domain theory. We want to support general recursive function definitions, but domain theory tells us that not all definitions are reasonable; some fail to be _continuous_ and thus represent unrealizable computations. To formalize an analogous notion of continuity for our non-termination monad, we write down the approximation relation on computation results that we have had in mind all along. *)
adam@352 467
adam@352 468 Section lattice.
adam@352 469 Variable A : Type.
adam@352 470
adam@352 471 Definition leq (x y : option A) :=
adam@352 472 forall v, x = Some v -> y = Some v.
adam@352 473 End lattice.
adam@352 474
adam@355 475 (** We now have the tools we need to define a new [Fix] combinator that, unlike the one we saw in the prior section, does not require a termination proof, and in fact admits recursive definition of functions that fail to terminate on some or all inputs. *)
adam@352 476
adam@352 477 Section Fix.
adam@355 478 (** First, we have the function domain and range types. *)
adam@355 479
adam@352 480 Variables A B : Type.
adam@355 481
adam@355 482 (** Next comes the function body, which is written as though it can be parameterized over itself, for recursive calls. *)
adam@355 483
adam@352 484 Variable f : (A -> computation B) -> (A -> computation B).
adam@352 485
adam@355 486 (** Finally, we impose an obligation to prove that the body [f] is continuous. That is, when [f] terminates according to one recursive version of itself, it also terminates with the same result at the same approximation level when passed a recursive version that refines the original, according to [leq]. *)
adam@355 487
adam@352 488 Hypothesis f_continuous : forall n v v1 x,
adam@352 489 runTo (f v1 x) n v
adam@352 490 -> forall (v2 : A -> computation B),
adam@352 491 (forall x, leq (proj1_sig (v1 x) n) (proj1_sig (v2 x) n))
adam@352 492 -> runTo (f v2 x) n v.
adam@352 493
adam@355 494 (** The computational part of the [Fix] combinator is easy to define. At approximation level 0, we diverge; at higher levels, we run the body with a functional argument drawn from the next lower level. *)
adam@355 495
adam@352 496 Fixpoint Fix' (n : nat) (x : A) : computation B :=
adam@352 497 match n with
adam@352 498 | O => Bottom _
adam@352 499 | S n' => f (Fix' n') x
adam@352 500 end.
adam@352 501
adam@355 502 (** Now it is straightforward to package [Fix'] as a computation combinator [Fix]. *)
adam@355 503
adam@352 504 Hint Extern 1 (_ >= _) => omega.
adam@352 505 Hint Unfold leq.
adam@352 506
adam@352 507 Lemma Fix'_ok : forall steps n x v, proj1_sig (Fix' n x) steps = Some v
adam@352 508 -> forall n', n' >= n
adam@352 509 -> proj1_sig (Fix' n' x) steps = Some v.
adam@352 510 unfold runTo in *; induction n; crush;
adam@352 511 match goal with
adam@352 512 | [ H : _ >= _ |- _ ] => inversion H; crush; eauto
adam@352 513 end.
adam@352 514 Qed.
adam@352 515
adam@352 516 Hint Resolve Fix'_ok.
adam@352 517
adam@352 518 Hint Extern 1 (proj1_sig _ _ = _) => simpl;
adam@352 519 match goal with
adam@352 520 | [ |- proj1_sig ?E _ = _ ] => eapply (proj2_sig E)
adam@352 521 end.
adam@352 522
adam@352 523 Definition Fix : A -> computation B.
adam@352 524 intro x; exists (fun n => proj1_sig (Fix' n x) n); abstract run.
adam@352 525 Defined.
adam@352 526
adam@355 527 (** Finally, we can prove that [Fix] obeys the expected computation rule. *)
adam@352 528
adam@352 529 Theorem run_Fix : forall x v,
adam@352 530 run (f Fix x) v
adam@352 531 -> run (Fix x) v.
adam@352 532 run; match goal with
adam@352 533 | [ n : nat |- _ ] => exists (S n); eauto
adam@352 534 end.
adam@352 535 Qed.
adam@352 536 End Fix.
adam@352 537
adam@355 538 (* begin hide *)
adam@352 539 Lemma leq_Some : forall A (x y : A), leq (Some x) (Some y)
adam@352 540 -> x = y.
adam@426 541 intros ? ? ? H; generalize (H _ (eq_refl _)); crush.
adam@352 542 Qed.
adam@352 543
adam@352 544 Lemma leq_None : forall A (x y : A), leq (Some x) None
adam@352 545 -> False.
adam@426 546 intros ? ? ? H; generalize (H _ (eq_refl _)); crush.
adam@352 547 Qed.
adam@352 548
adam@355 549 Ltac mergeSort' := run;
adam@355 550 repeat (match goal with
adam@355 551 | [ |- context[match ?E with O => _ | S _ => _ end] ] => destruct E
adam@355 552 end; run);
adam@355 553 repeat match goal with
adam@355 554 | [ H : forall x, leq (proj1_sig (?f x) _) (proj1_sig (?g x) _) |- _ ] =>
adam@355 555 match goal with
adam@355 556 | [ H1 : f ?arg = _, H2 : g ?arg = _ |- _ ] =>
adam@355 557 generalize (H arg); rewrite H1; rewrite H2; clear H1 H2; simpl; intro
adam@355 558 end
adam@355 559 end; run; repeat match goal with
adam@355 560 | [ H : _ |- _ ] => (apply leq_None in H; tauto) || (apply leq_Some in H; subst)
adam@355 561 end; auto.
adam@355 562 (* end hide *)
adam@355 563
adam@355 564 (** After all that work, it is now fairly painless to define a version of [mergeSort] that requires no proof of termination. We appeal to a program-specific tactic whose definition is hidden here but present in the book source. *)
adam@355 565
adam@352 566 Definition mergeSort' : forall A, (A -> A -> bool) -> list A -> computation (list A).
adam@352 567 refine (fun A le => Fix
adam@352 568 (fun (mergeSort : list A -> computation (list A))
adam@352 569 (ls : list A) =>
adam@352 570 if le_lt_dec 2 (length ls)
adam@352 571 then let lss := partition ls in
adam@352 572 ls1 <- mergeSort (fst lss);
adam@352 573 ls2 <- mergeSort (snd lss);
adam@352 574 Return (merge le ls1 ls2)
adam@355 575 else Return ls) _); abstract mergeSort'.
adam@352 576 Defined.
adam@352 577
adam@424 578 (** Furthermore, "running" [mergeSort'] on concrete inputs is as easy as choosing a sufficiently high approximation level and letting Coq's computation rules do the rest. Contrast this with the proof work that goes into deriving an evaluation fact for a deeply embedded language, with one explicit proof rule application per execution step. *)
adam@352 579
adam@352 580 Lemma test_mergeSort' : run (mergeSort' leb (1 :: 2 :: 36 :: 8 :: 19 :: nil))
adam@352 581 (1 :: 2 :: 8 :: 19 :: 36 :: nil).
adam@352 582 exists 4; reflexivity.
adam@352 583 Qed.
adam@352 584
adam@453 585 (** There is another benefit of our new [Fix] compared with the one we used in the previous section: we can now write recursive functions that sometimes fail to terminate, without losing easy reasoning principles for the terminating cases. Consider this simple example, which appeals to another tactic whose definition we elide here. *)
adam@355 586
adam@355 587 (* begin hide *)
adam@355 588 Ltac looper := unfold leq in *; run;
adam@355 589 repeat match goal with
adam@355 590 | [ x : unit |- _ ] => destruct x
adam@355 591 | [ x : bool |- _ ] => destruct x
adam@355 592 end; auto.
adam@355 593 (* end hide *)
adam@355 594
adam@352 595 Definition looper : bool -> computation unit.
adam@352 596 refine (Fix (fun looper (b : bool) =>
adam@355 597 if b then Return tt else looper b) _); abstract looper.
adam@352 598 Defined.
adam@352 599
adam@352 600 Lemma test_looper : run (looper true) tt.
adam@352 601 exists 1; reflexivity.
adam@352 602 Qed.
adam@354 603
adam@355 604 (** As before, proving outputs for specific inputs is as easy as demonstrating a high enough approximation level.
adam@355 605
adam@424 606 There are other theorems that are important to prove about combinators like [Return], [Bind], and [Fix]. In general, for a computation [c], we sometimes have a hypothesis proving [run c v] for some [v], and we want to perform inversion to deduce what [v] must be. Each combinator should ideally have a theorem of that kind, for [c] built directly from that combinator. We have omitted such theorems here, but they are not hard to prove. In general, the domain theory-inspired approach avoids the type-theoretic "gotchas" that tend to show up in approaches that try to mix normal Coq computation with explicit syntax types. The next section of this chapter demonstrates two alternate approaches of that sort. *)
adam@355 607
adam@354 608
adam@354 609 (** * Co-Inductive Non-Termination Monads *)
adam@354 610
adam@356 611 (** There are two key downsides to both of the previous approaches: both require unusual syntax based on explicit calls to fixpoint combinators, and both generate immediate proof obligations about the bodies of recursive definitions. In Chapter 5, we have already seen how co-inductive types support recursive definitions that exhibit certain well-behaved varieties of non-termination. It turns out that we can leverage that co-induction support for encoding of general recursive definitions, by adding layers of co-inductive syntax. In effect, we mix elements of shallow and deep embeddings.
adam@356 612
adam@356 613 Our first example of this kind, proposed by Capretta%~\cite{Capretta}%, defines a silly-looking type of thunks; that is, computations that may be forced to yield results, if they terminate. *)
adam@356 614
adam@354 615 CoInductive thunk (A : Type) : Type :=
adam@354 616 | Answer : A -> thunk A
adam@354 617 | Think : thunk A -> thunk A.
adam@354 618
adam@356 619 (** A computation is either an immediate [Answer] or another computation wrapped inside [Think]. Since [thunk] is co-inductive, every [thunk] type is inhabited by an infinite nesting of [Think]s, standing for non-termination. Terminating results are [Answer] wrapped inside some finite number of [Think]s.
adam@356 620
adam@424 621 Why bother to write such a strange definition? The definition of [thunk] is motivated by the ability it gives us to define a "bind" operation, similar to the one we defined in the previous section. *)
adam@356 622
adam@356 623 CoFixpoint TBind A B (m1 : thunk A) (m2 : A -> thunk B) : thunk B :=
adam@354 624 match m1 with
adam@354 625 | Answer x => m2 x
adam@354 626 | Think m1' => Think (TBind m1' m2)
adam@354 627 end.
adam@354 628
adam@356 629 (** Note that the definition would violate the co-recursion guardedness restriction if we left out the seemingly superfluous [Think] on the righthand side of the second [match] branch.
adam@356 630
adam@356 631 We can prove that [Answer] and [TBind] form a monad for [thunk]. The proof is omitted here but present in the book source. As usual for this sort of proof, a key element is choosing an appropriate notion of equality for [thunk]s. *)
adam@356 632
adam@356 633 (* begin hide *)
adam@354 634 CoInductive thunk_eq A : thunk A -> thunk A -> Prop :=
adam@354 635 | EqAnswer : forall x, thunk_eq (Answer x) (Answer x)
adam@354 636 | EqThinkL : forall m1 m2, thunk_eq m1 m2 -> thunk_eq (Think m1) m2
adam@354 637 | EqThinkR : forall m1 m2, thunk_eq m1 m2 -> thunk_eq m1 (Think m2).
adam@354 638
adam@354 639 Section thunk_eq_coind.
adam@354 640 Variable A : Type.
adam@354 641 Variable P : thunk A -> thunk A -> Prop.
adam@354 642
adam@354 643 Hypothesis H : forall m1 m2, P m1 m2
adam@354 644 -> match m1, m2 with
adam@354 645 | Answer x1, Answer x2 => x1 = x2
adam@354 646 | Think m1', Think m2' => P m1' m2'
adam@354 647 | Think m1', _ => P m1' m2
adam@354 648 | _, Think m2' => P m1 m2'
adam@354 649 end.
adam@354 650
adam@354 651 Theorem thunk_eq_coind : forall m1 m2, P m1 m2 -> thunk_eq m1 m2.
adam@354 652 cofix; intros;
adam@354 653 match goal with
adam@354 654 | [ H' : P _ _ |- _ ] => specialize (H H'); clear H'
adam@354 655 end; destruct m1; destruct m2; subst; repeat constructor; auto.
adam@354 656 Qed.
adam@354 657 End thunk_eq_coind.
adam@356 658 (* end hide *)
adam@356 659
adam@356 660 (** In the proofs to follow, we will need a function similar to one we saw in Chapter 5, to pull apart and reassemble a [thunk] in a way that provokes reduction of co-recursive calls. *)
adam@354 661
adam@354 662 Definition frob A (m : thunk A) : thunk A :=
adam@354 663 match m with
adam@354 664 | Answer x => Answer x
adam@354 665 | Think m' => Think m'
adam@354 666 end.
adam@354 667
adam@354 668 Theorem frob_eq : forall A (m : thunk A), frob m = m.
adam@354 669 destruct m; reflexivity.
adam@354 670 Qed.
adam@354 671
adam@356 672 (* begin hide *)
adam@354 673 Theorem thunk_eq_frob : forall A (m1 m2 : thunk A),
adam@354 674 thunk_eq (frob m1) (frob m2)
adam@354 675 -> thunk_eq m1 m2.
adam@354 676 intros; repeat rewrite frob_eq in *; auto.
adam@354 677 Qed.
adam@354 678
adam@354 679 Ltac findDestr := match goal with
adam@354 680 | [ |- context[match ?E with Answer _ => _ | Think _ => _ end] ] =>
adam@354 681 match E with
adam@354 682 | context[match _ with Answer _ => _ | Think _ => _ end] => fail 1
adam@354 683 | _ => destruct E
adam@354 684 end
adam@354 685 end.
adam@354 686
adam@354 687 Theorem thunk_eq_refl : forall A (m : thunk A), thunk_eq m m.
adam@354 688 intros; apply (thunk_eq_coind (fun m1 m2 => m1 = m2)); crush; findDestr; reflexivity.
adam@354 689 Qed.
adam@354 690
adam@354 691 Hint Resolve thunk_eq_refl.
adam@354 692
adam@354 693 Theorem tleft_identity : forall A B (a : A) (f : A -> thunk B),
adam@354 694 thunk_eq (TBind (Answer a) f) (f a).
adam@354 695 intros; apply thunk_eq_frob; crush.
adam@354 696 Qed.
adam@354 697
adam@354 698 Theorem tright_identity : forall A (m : thunk A),
adam@354 699 thunk_eq (TBind m (@Answer _)) m.
adam@354 700 intros; apply (thunk_eq_coind (fun m1 m2 => m1 = TBind m2 (@Answer _))); crush;
adam@354 701 findDestr; reflexivity.
adam@354 702 Qed.
adam@354 703
adam@354 704 Lemma TBind_Answer : forall (A B : Type) (v : A) (m2 : A -> thunk B),
adam@354 705 TBind (Answer v) m2 = m2 v.
adam@354 706 intros; rewrite <- (frob_eq (TBind (Answer v) m2));
adam@354 707 simpl; findDestr; reflexivity.
adam@354 708 Qed.
adam@354 709
adam@375 710 Hint Rewrite TBind_Answer.
adam@354 711
adam@355 712 (** printing exists $\exists$ *)
adam@355 713
adam@354 714 Theorem tassociativity : forall A B C (m : thunk A) (f : A -> thunk B) (g : B -> thunk C),
adam@354 715 thunk_eq (TBind (TBind m f) g) (TBind m (fun x => TBind (f x) g)).
adam@354 716 intros; apply (thunk_eq_coind (fun m1 m2 => (exists m,
adam@354 717 m1 = TBind (TBind m f) g
adam@354 718 /\ m2 = TBind m (fun x => TBind (f x) g))
adam@354 719 \/ m1 = m2)); crush; eauto; repeat (findDestr; crush; eauto).
adam@354 720 Qed.
adam@356 721 (* end hide *)
adam@356 722
adam@356 723 (** As a simple example, here is how we might define a tail-recursive factorial function. *)
adam@354 724
adam@354 725 CoFixpoint fact (n acc : nat) : thunk nat :=
adam@354 726 match n with
adam@354 727 | O => Answer acc
adam@354 728 | S n' => Think (fact n' (S n' * acc))
adam@354 729 end.
adam@354 730
adam@356 731 (** To test our definition, we need an evaluation relation that characterizes results of evaluating [thunk]s. *)
adam@356 732
adam@354 733 Inductive eval A : thunk A -> A -> Prop :=
adam@354 734 | EvalAnswer : forall x, eval (Answer x) x
adam@354 735 | EvalThink : forall m x, eval m x -> eval (Think m) x.
adam@354 736
adam@375 737 Hint Rewrite frob_eq.
adam@354 738
adam@354 739 Lemma eval_frob : forall A (c : thunk A) x,
adam@354 740 eval (frob c) x
adam@354 741 -> eval c x.
adam@354 742 crush.
adam@354 743 Qed.
adam@354 744
adam@354 745 Theorem eval_fact : eval (fact 5 1) 120.
adam@354 746 repeat (apply eval_frob; simpl; constructor).
adam@354 747 Qed.
adam@354 748
adam@356 749 (** We need to apply constructors of [eval] explicitly, but the process is easy to automate completely for concrete input programs.
adam@356 750
adam@465 751 Now consider another very similar definition, this time of a Fibonacci number function. *)
adam@357 752
adam@357 753 Notation "x <- m1 ; m2" :=
adam@357 754 (TBind m1 (fun x => m2)) (right associativity, at level 70).
adam@357 755
adam@404 756 (* begin hide *)
adam@437 757 (* begin thide *)
adam@424 758 Definition fib := pred.
adam@437 759 (* end thide *)
adam@404 760 (* end hide *)
adam@404 761
adam@453 762 (** %\vspace{-.3in}%[[
adam@354 763 CoFixpoint fib (n : nat) : thunk nat :=
adam@354 764 match n with
adam@354 765 | 0 => Answer 1
adam@354 766 | 1 => Answer 1
adam@357 767 | _ => n1 <- fib (pred n);
adam@357 768 n2 <- fib (pred (pred n));
adam@357 769 Answer (n1 + n2)
adam@354 770 end.
adam@354 771 ]]
adam@354 772
adam@356 773 Coq complains that the guardedness condition is violated. The two recursive calls are immediate arguments to [TBind], but [TBind] is not a constructor of [thunk]. Rather, it is a defined function. This example shows a very serious limitation of [thunk] for traditional functional programming: it is not, in general, possible to make recursive calls and then make further recursive calls, depending on the first call's result. The [fact] example succeeded because it was already tail recursive, meaning no further computation is needed after a recursive call.
adam@356 774
adam@356 775 %\medskip%
adam@356 776
adam@424 777 I know no easy fix for this problem of [thunk], but we can define an alternate co-inductive monad that avoids the problem, based on a proposal by Megacz%~\cite{Megacz}%. We ran into trouble because [TBind] was not a constructor of [thunk], so let us define a new type family where "bind" is a constructor. *)
adam@354 778
adam@354 779 CoInductive comp (A : Type) : Type :=
adam@354 780 | Ret : A -> comp A
adam@354 781 | Bnd : forall B, comp B -> (B -> comp A) -> comp A.
adam@354 782
adam@404 783 (** This example shows off Coq's support for%\index{recursively non-uniform parameters}% _recursively non-uniform parameters_, as in the case of the parameter [A] declared above, where each constructor's type ends in [comp A], but there is a recursive use of [comp] with a different parameter [B]. Beside that technical wrinkle, we see the simplest possible definition of a monad, via a type whose two constructors are precisely the monad operators.
adam@356 784
adam@356 785 It is easy to define the semantics of terminating [comp] computations. *)
adam@356 786
adam@354 787 Inductive exec A : comp A -> A -> Prop :=
adam@354 788 | ExecRet : forall x, exec (Ret x) x
adam@354 789 | ExecBnd : forall B (c : comp B) (f : B -> comp A) x1 x2, exec (A := B) c x1
adam@354 790 -> exec (f x1) x2
adam@354 791 -> exec (Bnd c f) x2.
adam@354 792
adam@356 793 (** We can also prove that [Ret] and [Bnd] form a monad according to a notion of [comp] equality based on [exec], but we omit details here; they are in the book source at this point. *)
adam@356 794
adam@356 795 (* begin hide *)
adam@354 796 Hint Constructors exec.
adam@354 797
adam@354 798 Definition comp_eq A (c1 c2 : comp A) := forall r, exec c1 r <-> exec c2 r.
adam@354 799
adam@354 800 Ltac inverter := repeat match goal with
adam@354 801 | [ H : exec _ _ |- _ ] => inversion H; []; crush
adam@354 802 end.
adam@354 803
adam@354 804 Theorem cleft_identity : forall A B (a : A) (f : A -> comp B),
adam@354 805 comp_eq (Bnd (Ret a) f) (f a).
adam@354 806 red; crush; inverter; eauto.
adam@354 807 Qed.
adam@354 808
adam@354 809 Theorem cright_identity : forall A (m : comp A),
adam@354 810 comp_eq (Bnd m (@Ret _)) m.
adam@354 811 red; crush; inverter; eauto.
adam@354 812 Qed.
adam@354 813
adam@354 814 Lemma cassociativity1 : forall A B C (f : A -> comp B) (g : B -> comp C) r c,
adam@354 815 exec c r
adam@354 816 -> forall m, c = Bnd (Bnd m f) g
adam@354 817 -> exec (Bnd m (fun x => Bnd (f x) g)) r.
adam@354 818 induction 1; crush.
adam@354 819 match goal with
adam@354 820 | [ H : Bnd _ _ = Bnd _ _ |- _ ] => injection H; clear H; intros; try subst
adam@354 821 end.
adam@354 822 move H3 after A.
adam@354 823 generalize dependent B0.
adam@354 824 do 2 intro.
adam@354 825 subst.
adam@354 826 crush.
adam@354 827 inversion H; clear H; crush.
adam@354 828 eauto.
adam@354 829 Qed.
adam@354 830
adam@354 831 Lemma cassociativity2 : forall A B C (f : A -> comp B) (g : B -> comp C) r c,
adam@354 832 exec c r
adam@354 833 -> forall m, c = Bnd m (fun x => Bnd (f x) g)
adam@354 834 -> exec (Bnd (Bnd m f) g) r.
adam@354 835 induction 1; crush.
adam@354 836 match goal with
adam@354 837 | [ H : Bnd _ _ = Bnd _ _ |- _ ] => injection H; clear H; intros; try subst
adam@354 838 end.
adam@354 839 move H3 after B.
adam@354 840 generalize dependent B0.
adam@354 841 do 2 intro.
adam@354 842 subst.
adam@354 843 crush.
adam@354 844 inversion H0; clear H0; crush.
adam@354 845 eauto.
adam@354 846 Qed.
adam@354 847
adam@354 848 Hint Resolve cassociativity1 cassociativity2.
adam@354 849
adam@354 850 Theorem cassociativity : forall A B C (m : comp A) (f : A -> comp B) (g : B -> comp C),
adam@354 851 comp_eq (Bnd (Bnd m f) g) (Bnd m (fun x => Bnd (f x) g)).
adam@354 852 red; crush; eauto.
adam@354 853 Qed.
adam@356 854 (* end hide *)
adam@356 855
adam@424 856 (** Not only can we define the Fibonacci function with the new monad, but even our running example of merge sort becomes definable. By shadowing our previous notation for "bind,", we can write almost exactly the same code as in our previous [mergeSort'] definition, but with less syntactic clutter. *)
adam@356 857
adam@356 858 Notation "x <- m1 ; m2" := (Bnd m1 (fun x => m2)).
adam@354 859
adam@354 860 CoFixpoint mergeSort'' A (le : A -> A -> bool) (ls : list A) : comp (list A) :=
adam@354 861 if le_lt_dec 2 (length ls)
adam@354 862 then let lss := partition ls in
adam@356 863 ls1 <- mergeSort'' le (fst lss);
adam@356 864 ls2 <- mergeSort'' le (snd lss);
adam@356 865 Ret (merge le ls1 ls2)
adam@354 866 else Ret ls.
adam@354 867
adam@356 868 (** To execute this function, we go through the usual exercise of writing a function to catalyze evaluation of co-recursive calls. *)
adam@356 869
adam@354 870 Definition frob' A (c : comp A) :=
adam@354 871 match c with
adam@354 872 | Ret x => Ret x
adam@354 873 | Bnd _ c' f => Bnd c' f
adam@354 874 end.
adam@354 875
adam@354 876 Lemma exec_frob : forall A (c : comp A) x,
adam@354 877 exec (frob' c) x
adam@354 878 -> exec c x.
adam@356 879 destruct c; crush.
adam@354 880 Qed.
adam@354 881
adam@356 882 (** Now the same sort of proof script that we applied for testing [thunk]s will get the job done. *)
adam@356 883
adam@354 884 Lemma test_mergeSort'' : exec (mergeSort'' leb (1 :: 2 :: 36 :: 8 :: 19 :: nil))
adam@354 885 (1 :: 2 :: 8 :: 19 :: 36 :: nil).
adam@354 886 repeat (apply exec_frob; simpl; econstructor).
adam@354 887 Qed.
adam@354 888
adam@356 889 (** Have we finally reached the ideal solution for encoding general recursive definitions, with minimal hassle in syntax and proof obligations? Unfortunately, we have not, as [comp] has a serious expressivity weakness. Consider the following definition of a curried addition function: *)
adam@356 890
adam@354 891 Definition curriedAdd (n : nat) := Ret (fun m : nat => Ret (n + m)).
adam@354 892
adam@356 893 (** This definition works fine, but we run into trouble when we try to apply it in a trivial way.
adam@356 894 [[
adam@356 895 Definition testCurriedAdd := Bnd (curriedAdd 2) (fun f => f 3).
adam@354 896 ]]
adam@354 897
adam@354 898 <<
adam@354 899 Error: Universe inconsistency.
adam@354 900 >>
adam@356 901
adam@356 902 The problem has to do with rules for inductive definitions that we still study in more detail in Chapter 12. Briefly, recall that the type of the constructor [Bnd] quantifies over a type [B]. To make [testCurriedAdd] work, we would need to instantiate [B] as [nat -> comp nat]. However, Coq enforces a %\emph{predicativity restriction}% that (roughly) no quantifier in an inductive or co-inductive type's definition may ever be instantiated with a term that contains the type being defined. Chapter 12 presents the exact mechanism by which this restriction is enforced, but for now our conclusion is that [comp] is fatally flawed as a way of encoding interesting higher-order functional programs that use general recursion. *)
adam@354 903
adam@354 904
adam@357 905 (** * Comparing the Alternatives *)
adam@354 906
adam@453 907 (** We have seen four different approaches to encoding general recursive definitions in Coq. Among them there is no clear champion that dominates the others in every important way. Instead, we close the chapter by comparing the techniques along a number of dimensions. Every technique allows recursive definitions with termination arguments that go beyond Coq's built-in termination checking, so we must turn to subtler points to highlight differences.
adam@356 908
adam@356 909 One useful property is automatic integration with normal Coq programming. That is, we would like the type of a function to be the same, whether or not that function is defined using an interesting recursion pattern. Only the first of the four techniques, well-founded recursion, meets this criterion. It is also the only one of the four to meet the related criterion that evaluation of function calls can take place entirely inside Coq's built-in computation machinery. The monad inspired by domain theory occupies some middle ground in this dimension, since generally standard computation is enough to evaluate a term once a high enough approximation level is provided.
adam@356 910
adam@356 911 Another useful property is that a function and its termination argument may be developed separately. We may even want to define functions that fail to terminate on some or all inputs. The well-founded recursion technique does not have this property, but the other three do.
adam@356 912
adam@356 913 One minor plus is the ability to write recursive definitions in natural syntax, rather than with calls to higher-order combinators. This downside of the first two techniques is actually rather easy to get around using Coq's notation mechanism, though we leave the details as an exercise for the reader.
adam@356 914
adam@465 915 The first two techniques impose proof obligations that are more basic than termination arguments, where well-founded recursion requires a proof of extensionality and domain-theoretic recursion requires a proof of continuity. A function may not be defined, and thus may not be computed with, until these obligations are proved. The co-inductive techniques avoid this problem, as recursive definitions may be made without any proof obligations.
adam@356 916
adam@356 917 We can also consider support for common idioms in functional programming. For instance, the [thunk] monad effectively only supports recursion that is tail recursion, while the others allow arbitrary recursion schemes.
adam@356 918
adam@356 919 On the other hand, the [comp] monad does not support the effective mixing of higher-order functions and general recursion, while all the other techniques do. For instance, we can finish the failed [curriedAdd] example in the domain-theoretic monad. *)
adam@356 920
adam@354 921 Definition curriedAdd' (n : nat) := Return (fun m : nat => Return (n + m)).
adam@354 922
adam@356 923 Definition testCurriedAdd := Bind (curriedAdd' 2) (fun f => f 3).
adam@356 924
adam@357 925 (** The same techniques also apply to more interesting higher-order functions like list map, and, as in all four techniques, we can mix primitive and general recursion, preferring the former when possible to avoid proof obligations. *)
adam@354 926
adam@354 927 Fixpoint map A B (f : A -> computation B) (ls : list A) : computation (list B) :=
adam@354 928 match ls with
adam@354 929 | nil => Return nil
adam@354 930 | x :: ls' => Bind (f x) (fun x' =>
adam@354 931 Bind (map f ls') (fun ls'' =>
adam@354 932 Return (x' :: ls'')))
adam@354 933 end.
adam@354 934
adam@355 935 (** remove printing exists *)
adam@356 936 Theorem test_map : run (map (fun x => Return (S x)) (1 :: 2 :: 3 :: nil))
adam@356 937 (2 :: 3 :: 4 :: nil).
adam@354 938 exists 1; reflexivity.
adam@354 939 Qed.
adam@356 940
adam@356 941 (** One further disadvantage of [comp] is that we cannot prove an inversion lemma for executions of [Bind] without appealing to an %\emph{axiom}%, a logical complication that we discuss at more length in Chapter 12. The other three techniques allow proof of all the important theorems within the normal logic of Coq.
adam@356 942
adam@357 943 Perhaps one theme of our comparison is that one must trade off between, on one hand, functional programming expressiveness and compatibility with normal Coq types and computation; and, on the other hand, the level of proof obligations one is willing to handle at function definition time. *)