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MIT 6.822
author Adam Chlipala <adam@chlipala.net>
date Mon, 18 Dec 2017 17:05:53 -0500
parents 16d701d4bd82
children af97676583f3
rev   line source
adam@534 1 (* Copyright (c) 2006, 2011-2012, 2015, 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@534 11 Require Import Arith List Omega.
adam@350 12
adam@534 13 Require Import Cpdt.CpdtTactics Cpdt.Coinductive.
adam@350 14
adam@350 15 Set Implicit Arguments.
adam@534 16 Set Asymmetric Patterns.
adam@350 17 (* end hide *)
adam@350 18
adam@350 19
adam@350 20 (** %\chapter{General Recursion}% *)
adam@350 21
adam@353 22 (** 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 23
adam@424 24 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 25
adam@353 26 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 27
adam@404 28 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 29
adam@351 30
adam@351 31 (** * Well-Founded Recursion *)
adam@351 32
adam@404 33 (** 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 34
adam@351 35 Section mergeSort.
adam@351 36 Variable A : Type.
adam@351 37 Variable le : A -> A -> bool.
adam@475 38
adam@424 39 (** We have a set equipped with some "less-than-or-equal-to" test. *)
adam@351 40
adam@351 41 (** A standard function inserts an element into a sorted list, preserving sortedness. *)
adam@351 42
adam@351 43 Fixpoint insert (x : A) (ls : list A) : list A :=
adam@351 44 match ls with
adam@351 45 | nil => x :: nil
adam@351 46 | h :: ls' =>
adam@351 47 if le x h
adam@351 48 then x :: ls
adam@351 49 else h :: insert x ls'
adam@351 50 end.
adam@351 51
adam@351 52 (** 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 53
adam@351 54 Fixpoint merge (ls1 ls2 : list A) : list A :=
adam@351 55 match ls1 with
adam@351 56 | nil => ls2
adam@351 57 | h :: ls' => insert h (merge ls' ls2)
adam@351 58 end.
adam@351 59
adam@480 60 (** The last helper function for classic merge sort is the one that follows, to split a list arbitrarily into two pieces of approximately equal length. *)
adam@351 61
adam@480 62 Fixpoint split (ls : list A) : list A * list A :=
adam@351 63 match ls with
adam@351 64 | nil => (nil, nil)
adam@351 65 | h :: nil => (h :: nil, nil)
adam@351 66 | h1 :: h2 :: ls' =>
adam@480 67 let (ls1, ls2) := split ls' in
adam@351 68 (h1 :: ls1, h2 :: ls2)
adam@351 69 end.
adam@351 70
adam@424 71 (** Now, let us try to write the final sorting function, using a natural number "[<=]" test [leb] from the standard library.
adam@351 72 [[
adam@351 73 Fixpoint mergeSort (ls : list A) : list A :=
adam@453 74 if leb (length ls) 1
adam@351 75 then ls
adam@480 76 else let lss := split ls in
adam@351 77 merge (mergeSort (fst lss)) (mergeSort (snd lss)).
adam@351 78 ]]
adam@351 79
adam@351 80 <<
adam@351 81 Recursive call to mergeSort has principal argument equal to
adam@480 82 "fst (split ls)" instead of a subterm of "ls".
adam@351 83 >>
adam@351 84
adam@351 85 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 86
adam@351 87 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 88
adam@351 89 Print well_founded.
adam@351 90 (** %\vspace{-.15in}% [[
adam@351 91 well_founded =
adam@351 92 fun (A : Type) (R : A -> A -> Prop) => forall a : A, Acc R a
adam@351 93 ]]
adam@351 94
adam@404 95 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 96
adam@424 97 (* begin hide *)
adam@437 98 (* begin thide *)
adam@424 99 Definition Acc_intro' := Acc_intro.
adam@437 100 (* end thide *)
adam@424 101 (* end hide *)
adam@424 102
adam@351 103 Print Acc.
adam@351 104 (** %\vspace{-.15in}% [[
adam@351 105 Inductive Acc (A : Type) (R : A -> A -> Prop) (x : A) : Prop :=
adam@351 106 Acc_intro : (forall y : A, R y x -> Acc R y) -> Acc R x
adam@351 107 ]]
adam@351 108
adam@424 109 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@351 110
adam@474 111 CoInductive infiniteDecreasingChain A (R : A -> A -> Prop) : stream A -> Prop :=
adam@474 112 | ChainCons : forall x y s, infiniteDecreasingChain R (Cons y s)
adam@351 113 -> R y x
adam@474 114 -> infiniteDecreasingChain R (Cons x (Cons y s)).
adam@351 115
adam@351 116 (** We can now prove that any accessible element cannot be the beginning of any infinite decreasing chain. *)
adam@351 117
adam@351 118 (* begin thide *)
adam@474 119 Lemma noBadChains' : forall A (R : A -> A -> Prop) x, Acc R x
adam@474 120 -> forall s, ~infiniteDecreasingChain R (Cons x s).
adam@351 121 induction 1; crush;
adam@351 122 match goal with
adam@474 123 | [ H : infiniteDecreasingChain _ _ |- _ ] => inversion H; eauto
adam@351 124 end.
adam@351 125 Qed.
adam@351 126
adam@351 127 (** From here, the absence of infinite decreasing chains in well-founded sets is immediate. *)
adam@351 128
adam@474 129 Theorem noBadChains : forall A (R : A -> A -> Prop), well_founded R
adam@474 130 -> forall s, ~infiniteDecreasingChain R s.
adam@474 131 destruct s; apply noBadChains'; auto.
adam@351 132 Qed.
adam@351 133 (* end thide *)
adam@351 134
adam@351 135 (** 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@351 136
adam@351 137 Check Fix.
adam@351 138 (** %\vspace{-.15in}%[[
adam@351 139 Fix
adam@351 140 : forall (A : Type) (R : A -> A -> Prop),
adam@351 141 well_founded R ->
adam@351 142 forall P : A -> Type,
adam@351 143 (forall x : A, (forall y : A, R y x -> P y) -> P x) ->
adam@351 144 forall x : A, P x
adam@351 145 ]]
adam@351 146
adam@351 147 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@351 148 [[
adam@351 149 forall x : A, (forall y : A, R y x -> P y) -> P x
adam@351 150 ]]
adam@351 151
adam@424 152 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@351 153
adam@353 154 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@351 155
adam@351 156 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@351 157
adam@351 158 Definition lengthOrder (ls1 ls2 : list A) :=
adam@351 159 length ls1 < length ls2.
adam@351 160
adam@353 161 (** 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@351 162
adam@351 163 Hint Constructors Acc.
adam@351 164
adam@351 165 Lemma lengthOrder_wf' : forall len, forall ls, length ls <= len -> Acc lengthOrder ls.
adam@351 166 unfold lengthOrder; induction len; crush.
adam@351 167 Defined.
adam@351 168
adam@351 169 Theorem lengthOrder_wf : well_founded lengthOrder.
adam@351 170 red; intro; eapply lengthOrder_wf'; eauto.
adam@351 171 Defined.
adam@351 172
adam@514 173 (** Notice that we end these proofs with %\index{Vernacular commands!Defined}%[Defined], not [Qed]. Recall that [Defined] marks the theorems as %\emph{%#<i>#transparent#</i>#%}%, 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@351 174
adam@480 175 To justify our two recursive [mergeSort] calls, we will also need to prove that [split] respects the [lengthOrder] relation. These proofs, too, must be kept transparent, to avoid stuckness of [Fix] evaluation. We use the syntax [@foo] to reference identifier [foo] with its implicit argument behavior turned off. (The proof details below use Ltac features not introduced yet, and they are safe to skip for now.) *)
adam@351 176
adam@480 177 Lemma split_wf : forall len ls, 2 <= length ls <= len
adam@480 178 -> let (ls1, ls2) := split ls in
adam@351 179 lengthOrder ls1 ls /\ lengthOrder ls2 ls.
adam@351 180 unfold lengthOrder; induction len; crush; do 2 (destruct ls; crush);
adam@351 181 destruct (le_lt_dec 2 (length ls));
adam@351 182 repeat (match goal with
adam@351 183 | [ _ : length ?E < 2 |- _ ] => destruct E
adam@351 184 | [ _ : S (length ?E) < 2 |- _ ] => destruct E
adam@480 185 | [ IH : _ |- context[split ?L] ] =>
adam@480 186 specialize (IH L); destruct (split L); destruct IH
adam@351 187 end; crush).
adam@351 188 Defined.
adam@351 189
adam@480 190 Ltac split_wf := intros ls ?; intros; generalize (@split_wf (length ls) ls);
adam@480 191 destruct (split ls); destruct 1; crush.
adam@351 192
adam@480 193 Lemma split_wf1 : forall ls, 2 <= length ls
adam@480 194 -> lengthOrder (fst (split ls)) ls.
adam@480 195 split_wf.
adam@351 196 Defined.
adam@351 197
adam@480 198 Lemma split_wf2 : forall ls, 2 <= length ls
adam@480 199 -> lengthOrder (snd (split ls)) ls.
adam@480 200 split_wf.
adam@351 201 Defined.
adam@351 202
adam@480 203 Hint Resolve split_wf1 split_wf2.
adam@351 204
adam@453 205 (** 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@351 206
adam@351 207 Definition mergeSort : list A -> list A.
adam@351 208 (* begin thide *)
adam@351 209 refine (Fix lengthOrder_wf (fun _ => list A)
adam@351 210 (fun (ls : list A)
adam@351 211 (mergeSort : forall ls' : list A, lengthOrder ls' ls -> list A) =>
adam@351 212 if le_lt_dec 2 (length ls)
adam@480 213 then let lss := split ls in
adam@351 214 merge (mergeSort (fst lss) _) (mergeSort (snd lss) _)
adam@351 215 else ls)); subst lss; eauto.
adam@351 216 Defined.
adam@351 217 (* end thide *)
adam@351 218 End mergeSort.
adam@351 219
adam@351 220 (** The important thing is that it is now easy to evaluate calls to [mergeSort]. *)
adam@351 221
adam@351 222 Eval compute in mergeSort leb (1 :: 2 :: 36 :: 8 :: 19 :: nil).
adam@351 223 (** [= 1 :: 2 :: 8 :: 19 :: 36 :: nil] *)
adam@351 224
adam@441 225 (** %\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@351 226
adam@351 227 (* begin thide *)
adam@351 228 Theorem mergeSort_eq : forall A (le : A -> A -> bool) ls,
adam@351 229 mergeSort le ls = if le_lt_dec 2 (length ls)
adam@480 230 then let lss := split ls in
adam@351 231 merge le (mergeSort le (fst lss)) (mergeSort le (snd lss))
adam@351 232 else ls.
adam@351 233 intros; apply (Fix_eq (@lengthOrder_wf A) (fun _ => list A)); intros.
adam@351 234
adam@424 235 (** 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@351 236
adam@351 237 Check Fix_eq.
adam@351 238 (** %\vspace{-.15in}%[[
adam@351 239 Fix_eq
adam@351 240 : forall (A : Type) (R : A -> A -> Prop) (Rwf : well_founded R)
adam@351 241 (P : A -> Type)
adam@351 242 (F : forall x : A, (forall y : A, R y x -> P y) -> P x),
adam@351 243 (forall (x : A) (f g : forall y : A, R y x -> P y),
adam@351 244 (forall (y : A) (p : R y x), f y p = g y p) -> F x f = F x g) ->
adam@351 245 forall x : A,
adam@351 246 Fix Rwf P F x = F x (fun (y : A) (_ : R y x) => Fix Rwf P F y)
adam@351 247 ]]
adam@351 248
adam@465 249 Most such obligations are dischargeable with straightforward proof automation, and this example is no exception. *)
adam@351 250
adam@351 251 match goal with
adam@351 252 | [ |- context[match ?E with left _ => _ | right _ => _ end] ] => destruct E
adam@351 253 end; simpl; f_equal; auto.
adam@351 254 Qed.
adam@351 255 (* end thide *)
adam@351 256
adam@351 257 (** As a final test of our definition's suitability, we can extract to OCaml. *)
adam@351 258
adam@351 259 Extraction mergeSort.
adam@351 260
adam@351 261 (** <<
adam@351 262 let rec mergeSort le x =
adam@351 263 match le_lt_dec (S (S O)) (length x) with
adam@351 264 | Left ->
adam@480 265 let lss = split x in
adam@351 266 merge le (mergeSort le (fst lss)) (mergeSort le (snd lss))
adam@351 267 | Right -> x
adam@351 268 >>
adam@351 269
adam@353 270 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@351 271
adam@351 272 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@351 273
adam@351 274 Check well_founded_induction.
adam@351 275 (** %\vspace{-.15in}%[[
adam@351 276 well_founded_induction
adam@351 277 : forall (A : Type) (R : A -> A -> Prop),
adam@351 278 well_founded R ->
adam@351 279 forall P : A -> Set,
adam@351 280 (forall x : A, (forall y : A, R y x -> P y) -> P x) ->
adam@351 281 forall a : A, P a
adam@351 282 ]]
adam@351 283
adam@351 284 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@352 285
adam@352 286
adam@354 287 (** * A Non-Termination Monad Inspired by Domain Theory *)
adam@352 288
adam@424 289 (** 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@355 290
adam@355 291 Consider this definition of a type of computations. *)
adam@355 292
adam@352 293 Section computation.
adam@352 294 Variable A : Type.
adam@355 295 (** The type [A] describes the result a computation will yield, if it terminates.
adam@355 296
adam@355 297 We give a rich dependent type to computations themselves: *)
adam@352 298
adam@352 299 Definition computation :=
adam@352 300 {f : nat -> option A
adam@352 301 | forall (n : nat) (v : A),
adam@352 302 f n = Some v
adam@352 303 -> forall (n' : nat), n' >= n
adam@352 304 -> f n' = Some v}.
adam@352 305
adam@474 306 (** 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 subset 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@355 307
adam@355 308 It is easy to define a relation characterizing when a computation runs to a particular result at a particular approximation level. *)
adam@355 309
adam@352 310 Definition runTo (m : computation) (n : nat) (v : A) :=
adam@352 311 proj1_sig m n = Some v.
adam@352 312
adam@355 313 (** On top of [runTo], we also define [run], which is the most abstract notion of when a computation runs to a value. *)
adam@355 314
adam@352 315 Definition run (m : computation) (v : A) :=
adam@352 316 exists n, runTo m n v.
adam@352 317 End computation.
adam@352 318
adam@355 319 (** 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@355 320 (* begin hide *)
adam@355 321
adam@352 322 Hint Unfold runTo.
adam@352 323
adam@352 324 Ltac run' := unfold run, runTo in *; try red; crush;
adam@352 325 repeat (match goal with
adam@352 326 | [ _ : proj1_sig ?E _ = _ |- _ ] =>
adam@352 327 match goal with
adam@352 328 | [ x : _ |- _ ] =>
adam@352 329 match x with
adam@352 330 | E => destruct E
adam@352 331 end
adam@352 332 end
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 *; try subst
adam@352 335 | [ _ : context[match ?M with exist _ _ => _ end] |- _ ] => let Heq := fresh "Heq" in
adam@352 336 case_eq M; intros ? ? Heq; try rewrite Heq in *; subst
adam@352 337 | [ H : forall n v, ?E n = Some v -> _,
adam@352 338 _ : context[match ?E ?N with Some _ => _ | None => _ end] |- _ ] =>
adam@426 339 specialize (H N); destruct (E N); try rewrite (H _ (eq_refl _)) by auto; try discriminate
adam@352 340 | [ H : forall n v, ?E n = Some v -> _, H' : ?E _ = Some _ |- _ ] => rewrite (H _ _ H') by auto
adam@352 341 end; simpl in *); eauto 7.
adam@352 342
adam@352 343 Ltac run := run'; repeat (match goal with
adam@352 344 | [ H : forall n v, ?E n = Some v -> _
adam@352 345 |- context[match ?E ?N with Some _ => _ | None => _ end] ] =>
adam@426 346 specialize (H N); destruct (E N); try rewrite (H _ (eq_refl _)) by auto; try discriminate
adam@352 347 end; run').
adam@352 348
adam@352 349 Lemma ex_irrelevant : forall P : Prop, P -> exists n : nat, P.
adam@352 350 exists 0; auto.
adam@352 351 Qed.
adam@352 352
adam@352 353 Hint Resolve ex_irrelevant.
adam@352 354
adam@352 355 Require Import Max.
adam@352 356
adam@380 357 Theorem max_spec_le : forall n m, n <= m /\ max n m = m \/ m <= n /\ max n m = n.
adam@380 358 induction n; destruct m; simpl; intuition;
adam@380 359 specialize (IHn m); intuition.
adam@380 360 Qed.
adam@380 361
adam@352 362 Ltac max := intros n m; generalize (max_spec_le n m); crush.
adam@352 363
adam@352 364 Lemma max_1 : forall n m, max n m >= n.
adam@352 365 max.
adam@352 366 Qed.
adam@352 367
adam@352 368 Lemma max_2 : forall n m, max n m >= m.
adam@352 369 max.
adam@352 370 Qed.
adam@352 371
adam@352 372 Hint Resolve max_1 max_2.
adam@352 373
adam@352 374 Lemma ge_refl : forall n, n >= n.
adam@352 375 crush.
adam@352 376 Qed.
adam@352 377
adam@352 378 Hint Resolve ge_refl.
adam@352 379
adam@352 380 Hint Extern 1 => match goal with
adam@352 381 | [ H : _ = exist _ _ _ |- _ ] => rewrite H
adam@352 382 end.
adam@355 383 (* end hide *)
adam@355 384 (** remove printing exists *)
adam@355 385
adam@480 386 (** 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 #<tt>#%\coqdocvar{%run%}%#</tt># tactic. In contrast to the previous section, opaque proofs are fine here, since the proof components of computations do not influence evaluation behavior. It is generally preferable to make proofs opaque when possible, as this enforces a kind of modularity in the code to follow, preventing it from depending on any details of the proof. *)
adam@352 387
adam@352 388 Section Bottom.
adam@352 389 Variable A : Type.
adam@352 390
adam@352 391 Definition Bottom : computation A.
adam@352 392 exists (fun _ : nat => @None A); abstract run.
adam@352 393 Defined.
adam@352 394
adam@352 395 Theorem run_Bottom : forall v, ~run Bottom v.
adam@352 396 run.
adam@352 397 Qed.
adam@352 398 End Bottom.
adam@352 399
adam@355 400 (** A slightly more complicated example is [Return], which gives the same terminating answer at every approximation level. *)
adam@355 401
adam@352 402 Section Return.
adam@352 403 Variable A : Type.
adam@352 404 Variable v : A.
adam@352 405
adam@352 406 Definition Return : computation A.
adam@352 407 intros; exists (fun _ : nat => Some v); abstract run.
adam@352 408 Defined.
adam@352 409
adam@352 410 Theorem run_Return : run Return v.
adam@352 411 run.
adam@352 412 Qed.
adam@352 413 End Return.
adam@352 414
adam@474 415 (** 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. We implement bind using the notation [let (x, y) := e1 in e2], for pulling apart the value [e1] which may be thought of as a pair. The second component of a [computation] is a proof, which we do not need to mention directly in the definition of [Bind]. *)
adam@352 416
adam@352 417 Section Bind.
adam@352 418 Variables A B : Type.
adam@352 419 Variable m1 : computation A.
adam@352 420 Variable m2 : A -> computation B.
adam@352 421
adam@352 422 Definition Bind : computation B.
adam@352 423 exists (fun n =>
adam@357 424 let (f1, _) := m1 in
adam@352 425 match f1 n with
adam@352 426 | None => None
adam@352 427 | Some v =>
adam@357 428 let (f2, _) := m2 v in
adam@352 429 f2 n
adam@352 430 end); abstract run.
adam@352 431 Defined.
adam@352 432
adam@352 433 Theorem run_Bind : forall (v1 : A) (v2 : B),
adam@352 434 run m1 v1
adam@352 435 -> run (m2 v1) v2
adam@352 436 -> run Bind v2.
adam@352 437 run; match goal with
adam@352 438 | [ x : nat, y : nat |- _ ] => exists (max x y)
adam@352 439 end; run.
adam@352 440 Qed.
adam@352 441 End Bind.
adam@352 442
adam@355 443 (** A simple notation lets us write [Bind] calls the way they appear in Haskell. *)
adam@352 444
adam@352 445 Notation "x <- m1 ; m2" :=
adam@352 446 (Bind m1 (fun x => m2)) (right associativity, at level 70).
adam@352 447
adam@424 448 (** 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@355 449
adam@352 450 Definition meq A (m1 m2 : computation A) := forall n, proj1_sig m1 n = proj1_sig m2 n.
adam@352 451
adam@352 452 Theorem left_identity : forall A B (a : A) (f : A -> computation B),
adam@352 453 meq (Bind (Return a) f) (f a).
adam@352 454 run.
adam@352 455 Qed.
adam@352 456
adam@352 457 Theorem right_identity : forall A (m : computation A),
adam@352 458 meq (Bind m (@Return _)) m.
adam@352 459 run.
adam@352 460 Qed.
adam@352 461
adam@357 462 Theorem associativity : forall A B C (m : computation A)
adam@357 463 (f : A -> computation B) (g : B -> computation C),
adam@352 464 meq (Bind (Bind m f) g) (Bind m (fun x => Bind (f x) g)).
adam@352 465 run.
adam@352 466 Qed.
adam@352 467
adam@398 468 (** 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@352 469
adam@352 470 Section lattice.
adam@352 471 Variable A : Type.
adam@352 472
adam@352 473 Definition leq (x y : option A) :=
adam@352 474 forall v, x = Some v -> y = Some v.
adam@352 475 End lattice.
adam@352 476
adam@355 477 (** 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@352 478
adam@352 479 Section Fix.
adam@475 480
adam@355 481 (** First, we have the function domain and range types. *)
adam@355 482
adam@352 483 Variables A B : Type.
adam@355 484
adam@355 485 (** Next comes the function body, which is written as though it can be parameterized over itself, for recursive calls. *)
adam@355 486
adam@352 487 Variable f : (A -> computation B) -> (A -> computation B).
adam@352 488
adam@355 489 (** 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@355 490
adam@352 491 Hypothesis f_continuous : forall n v v1 x,
adam@352 492 runTo (f v1 x) n v
adam@352 493 -> forall (v2 : A -> computation B),
adam@352 494 (forall x, leq (proj1_sig (v1 x) n) (proj1_sig (v2 x) n))
adam@352 495 -> runTo (f v2 x) n v.
adam@352 496
adam@355 497 (** 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@355 498
adam@352 499 Fixpoint Fix' (n : nat) (x : A) : computation B :=
adam@352 500 match n with
adam@352 501 | O => Bottom _
adam@352 502 | S n' => f (Fix' n') x
adam@352 503 end.
adam@352 504
adam@355 505 (** Now it is straightforward to package [Fix'] as a computation combinator [Fix]. *)
adam@355 506
adam@352 507 Hint Extern 1 (_ >= _) => omega.
adam@352 508 Hint Unfold leq.
adam@352 509
adam@352 510 Lemma Fix'_ok : forall steps n x v, proj1_sig (Fix' n x) steps = Some v
adam@352 511 -> forall n', n' >= n
adam@352 512 -> proj1_sig (Fix' n' x) steps = Some v.
adam@352 513 unfold runTo in *; induction n; crush;
adam@352 514 match goal with
adam@352 515 | [ H : _ >= _ |- _ ] => inversion H; crush; eauto
adam@352 516 end.
adam@352 517 Qed.
adam@352 518
adam@352 519 Hint Resolve Fix'_ok.
adam@352 520
adam@352 521 Hint Extern 1 (proj1_sig _ _ = _) => simpl;
adam@352 522 match goal with
adam@352 523 | [ |- proj1_sig ?E _ = _ ] => eapply (proj2_sig E)
adam@352 524 end.
adam@352 525
adam@352 526 Definition Fix : A -> computation B.
adam@352 527 intro x; exists (fun n => proj1_sig (Fix' n x) n); abstract run.
adam@352 528 Defined.
adam@352 529
adam@355 530 (** Finally, we can prove that [Fix] obeys the expected computation rule. *)
adam@352 531
adam@352 532 Theorem run_Fix : forall x v,
adam@352 533 run (f Fix x) v
adam@352 534 -> run (Fix x) v.
adam@352 535 run; match goal with
adam@352 536 | [ n : nat |- _ ] => exists (S n); eauto
adam@352 537 end.
adam@352 538 Qed.
adam@352 539 End Fix.
adam@352 540
adam@355 541 (* begin hide *)
adam@352 542 Lemma leq_Some : forall A (x y : A), leq (Some x) (Some y)
adam@352 543 -> x = y.
adam@426 544 intros ? ? ? H; generalize (H _ (eq_refl _)); crush.
adam@352 545 Qed.
adam@352 546
adam@352 547 Lemma leq_None : forall A (x y : A), leq (Some x) None
adam@352 548 -> False.
adam@426 549 intros ? ? ? H; generalize (H _ (eq_refl _)); crush.
adam@352 550 Qed.
adam@352 551
adam@355 552 Ltac mergeSort' := run;
adam@355 553 repeat (match goal with
adam@355 554 | [ |- context[match ?E with O => _ | S _ => _ end] ] => destruct E
adam@355 555 end; run);
adam@355 556 repeat match goal with
adam@355 557 | [ H : forall x, leq (proj1_sig (?f x) _) (proj1_sig (?g x) _) |- _ ] =>
adam@355 558 match goal with
adam@355 559 | [ H1 : f ?arg = _, H2 : g ?arg = _ |- _ ] =>
adam@355 560 generalize (H arg); rewrite H1; rewrite H2; clear H1 H2; simpl; intro
adam@355 561 end
adam@355 562 end; run; repeat match goal with
adam@355 563 | [ H : _ |- _ ] => (apply leq_None in H; tauto) || (apply leq_Some in H; subst)
adam@355 564 end; auto.
adam@355 565 (* end hide *)
adam@355 566
adam@355 567 (** 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@355 568
adam@352 569 Definition mergeSort' : forall A, (A -> A -> bool) -> list A -> computation (list A).
adam@352 570 refine (fun A le => Fix
adam@352 571 (fun (mergeSort : list A -> computation (list A))
adam@352 572 (ls : list A) =>
adam@352 573 if le_lt_dec 2 (length ls)
adam@480 574 then let lss := split ls in
adam@352 575 ls1 <- mergeSort (fst lss);
adam@352 576 ls2 <- mergeSort (snd lss);
adam@352 577 Return (merge le ls1 ls2)
adam@355 578 else Return ls) _); abstract mergeSort'.
adam@352 579 Defined.
adam@352 580
adam@424 581 (** 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@352 582
adam@352 583 Lemma test_mergeSort' : run (mergeSort' leb (1 :: 2 :: 36 :: 8 :: 19 :: nil))
adam@352 584 (1 :: 2 :: 8 :: 19 :: 36 :: nil).
adam@352 585 exists 4; reflexivity.
adam@352 586 Qed.
adam@352 587
adam@453 588 (** 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@355 589
adam@355 590 (* begin hide *)
adam@355 591 Ltac looper := unfold leq in *; run;
adam@355 592 repeat match goal with
adam@355 593 | [ x : unit |- _ ] => destruct x
adam@355 594 | [ x : bool |- _ ] => destruct x
adam@355 595 end; auto.
adam@355 596 (* end hide *)
adam@355 597
adam@352 598 Definition looper : bool -> computation unit.
adam@352 599 refine (Fix (fun looper (b : bool) =>
adam@355 600 if b then Return tt else looper b) _); abstract looper.
adam@352 601 Defined.
adam@352 602
adam@352 603 Lemma test_looper : run (looper true) tt.
adam@352 604 exists 1; reflexivity.
adam@352 605 Qed.
adam@354 606
adam@355 607 (** As before, proving outputs for specific inputs is as easy as demonstrating a high enough approximation level.
adam@355 608
adam@480 609 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. In the final section of the chapter, we review the pros and cons of the different choices, coming to the conclusion that none of them is obviously better than any one of the others for all situations. *)
adam@355 610
adam@354 611
adam@354 612 (** * Co-Inductive Non-Termination Monads *)
adam@354 613
adam@356 614 (** 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@356 615
adam@356 616 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@356 617
adam@354 618 CoInductive thunk (A : Type) : Type :=
adam@354 619 | Answer : A -> thunk A
adam@354 620 | Think : thunk A -> thunk A.
adam@354 621
adam@356 622 (** 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@356 623
adam@424 624 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@356 625
adam@356 626 CoFixpoint TBind A B (m1 : thunk A) (m2 : A -> thunk B) : thunk B :=
adam@354 627 match m1 with
adam@354 628 | Answer x => m2 x
adam@354 629 | Think m1' => Think (TBind m1' m2)
adam@354 630 end.
adam@354 631
adam@356 632 (** 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@356 633
adam@356 634 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@356 635
adam@356 636 (* begin hide *)
adam@354 637 CoInductive thunk_eq A : thunk A -> thunk A -> Prop :=
adam@354 638 | EqAnswer : forall x, thunk_eq (Answer x) (Answer x)
adam@354 639 | EqThinkL : forall m1 m2, thunk_eq m1 m2 -> thunk_eq (Think m1) m2
adam@354 640 | EqThinkR : forall m1 m2, thunk_eq m1 m2 -> thunk_eq m1 (Think m2).
adam@354 641
adam@354 642 Section thunk_eq_coind.
adam@354 643 Variable A : Type.
adam@354 644 Variable P : thunk A -> thunk A -> Prop.
adam@354 645
adam@354 646 Hypothesis H : forall m1 m2, P m1 m2
adam@354 647 -> match m1, m2 with
adam@354 648 | Answer x1, Answer x2 => x1 = x2
adam@354 649 | Think m1', Think m2' => P m1' m2'
adam@354 650 | Think m1', _ => P m1' m2
adam@354 651 | _, Think m2' => P m1 m2'
adam@354 652 end.
adam@354 653
adam@354 654 Theorem thunk_eq_coind : forall m1 m2, P m1 m2 -> thunk_eq m1 m2.
adam@354 655 cofix; intros;
adam@354 656 match goal with
adam@354 657 | [ H' : P _ _ |- _ ] => specialize (H H'); clear H'
adam@354 658 end; destruct m1; destruct m2; subst; repeat constructor; auto.
adam@354 659 Qed.
adam@354 660 End thunk_eq_coind.
adam@356 661 (* end hide *)
adam@356 662
adam@356 663 (** 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@354 664
adam@354 665 Definition frob A (m : thunk A) : thunk A :=
adam@354 666 match m with
adam@354 667 | Answer x => Answer x
adam@354 668 | Think m' => Think m'
adam@354 669 end.
adam@354 670
adam@354 671 Theorem frob_eq : forall A (m : thunk A), frob m = m.
adam@354 672 destruct m; reflexivity.
adam@354 673 Qed.
adam@354 674
adam@356 675 (* begin hide *)
adam@354 676 Theorem thunk_eq_frob : forall A (m1 m2 : thunk A),
adam@354 677 thunk_eq (frob m1) (frob m2)
adam@354 678 -> thunk_eq m1 m2.
adam@354 679 intros; repeat rewrite frob_eq in *; auto.
adam@354 680 Qed.
adam@354 681
adam@354 682 Ltac findDestr := match goal with
adam@354 683 | [ |- context[match ?E with Answer _ => _ | Think _ => _ end] ] =>
adam@354 684 match E with
adam@354 685 | context[match _ with Answer _ => _ | Think _ => _ end] => fail 1
adam@354 686 | _ => destruct E
adam@354 687 end
adam@354 688 end.
adam@354 689
adam@354 690 Theorem thunk_eq_refl : forall A (m : thunk A), thunk_eq m m.
adam@354 691 intros; apply (thunk_eq_coind (fun m1 m2 => m1 = m2)); crush; findDestr; reflexivity.
adam@354 692 Qed.
adam@354 693
adam@354 694 Hint Resolve thunk_eq_refl.
adam@354 695
adam@354 696 Theorem tleft_identity : forall A B (a : A) (f : A -> thunk B),
adam@354 697 thunk_eq (TBind (Answer a) f) (f a).
adam@354 698 intros; apply thunk_eq_frob; crush.
adam@354 699 Qed.
adam@354 700
adam@354 701 Theorem tright_identity : forall A (m : thunk A),
adam@354 702 thunk_eq (TBind m (@Answer _)) m.
adam@354 703 intros; apply (thunk_eq_coind (fun m1 m2 => m1 = TBind m2 (@Answer _))); crush;
adam@354 704 findDestr; reflexivity.
adam@354 705 Qed.
adam@354 706
adam@354 707 Lemma TBind_Answer : forall (A B : Type) (v : A) (m2 : A -> thunk B),
adam@354 708 TBind (Answer v) m2 = m2 v.
adam@354 709 intros; rewrite <- (frob_eq (TBind (Answer v) m2));
adam@354 710 simpl; findDestr; reflexivity.
adam@354 711 Qed.
adam@354 712
adam@375 713 Hint Rewrite TBind_Answer.
adam@354 714
adam@355 715 (** printing exists $\exists$ *)
adam@355 716
adam@354 717 Theorem tassociativity : forall A B C (m : thunk A) (f : A -> thunk B) (g : B -> thunk C),
adam@354 718 thunk_eq (TBind (TBind m f) g) (TBind m (fun x => TBind (f x) g)).
adam@354 719 intros; apply (thunk_eq_coind (fun m1 m2 => (exists m,
adam@354 720 m1 = TBind (TBind m f) g
adam@354 721 /\ m2 = TBind m (fun x => TBind (f x) g))
adam@354 722 \/ m1 = m2)); crush; eauto; repeat (findDestr; crush; eauto).
adam@354 723 Qed.
adam@356 724 (* end hide *)
adam@356 725
adam@356 726 (** As a simple example, here is how we might define a tail-recursive factorial function. *)
adam@354 727
adam@354 728 CoFixpoint fact (n acc : nat) : thunk nat :=
adam@354 729 match n with
adam@354 730 | O => Answer acc
adam@354 731 | S n' => Think (fact n' (S n' * acc))
adam@354 732 end.
adam@354 733
adam@356 734 (** To test our definition, we need an evaluation relation that characterizes results of evaluating [thunk]s. *)
adam@356 735
adam@354 736 Inductive eval A : thunk A -> A -> Prop :=
adam@354 737 | EvalAnswer : forall x, eval (Answer x) x
adam@354 738 | EvalThink : forall m x, eval m x -> eval (Think m) x.
adam@354 739
adam@375 740 Hint Rewrite frob_eq.
adam@354 741
adam@354 742 Lemma eval_frob : forall A (c : thunk A) x,
adam@354 743 eval (frob c) x
adam@354 744 -> eval c x.
adam@354 745 crush.
adam@354 746 Qed.
adam@354 747
adam@354 748 Theorem eval_fact : eval (fact 5 1) 120.
adam@354 749 repeat (apply eval_frob; simpl; constructor).
adam@354 750 Qed.
adam@354 751
adam@356 752 (** We need to apply constructors of [eval] explicitly, but the process is easy to automate completely for concrete input programs.
adam@356 753
adam@465 754 Now consider another very similar definition, this time of a Fibonacci number function. *)
adam@357 755
adam@357 756 Notation "x <- m1 ; m2" :=
adam@357 757 (TBind m1 (fun x => m2)) (right associativity, at level 70).
adam@357 758
adam@404 759 (* begin hide *)
adam@437 760 (* begin thide *)
adam@424 761 Definition fib := pred.
adam@437 762 (* end thide *)
adam@404 763 (* end hide *)
adam@404 764
adam@453 765 (** %\vspace{-.3in}%[[
adam@354 766 CoFixpoint fib (n : nat) : thunk nat :=
adam@354 767 match n with
adam@354 768 | 0 => Answer 1
adam@354 769 | 1 => Answer 1
adam@357 770 | _ => n1 <- fib (pred n);
adam@357 771 n2 <- fib (pred (pred n));
adam@357 772 Answer (n1 + n2)
adam@354 773 end.
adam@354 774 ]]
adam@354 775
adam@356 776 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@356 777
adam@356 778 %\medskip%
adam@356 779
adam@424 780 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@354 781
adam@354 782 CoInductive comp (A : Type) : Type :=
adam@354 783 | Ret : A -> comp A
adam@354 784 | Bnd : forall B, comp B -> (B -> comp A) -> comp A.
adam@354 785
adam@404 786 (** 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@356 787
adam@356 788 It is easy to define the semantics of terminating [comp] computations. *)
adam@356 789
adam@354 790 Inductive exec A : comp A -> A -> Prop :=
adam@354 791 | ExecRet : forall x, exec (Ret x) x
adam@354 792 | ExecBnd : forall B (c : comp B) (f : B -> comp A) x1 x2, exec (A := B) c x1
adam@354 793 -> exec (f x1) x2
adam@354 794 -> exec (Bnd c f) x2.
adam@354 795
adam@356 796 (** 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@356 797
adam@356 798 (* begin hide *)
adam@354 799 Hint Constructors exec.
adam@354 800
adam@354 801 Definition comp_eq A (c1 c2 : comp A) := forall r, exec c1 r <-> exec c2 r.
adam@354 802
adam@354 803 Ltac inverter := repeat match goal with
adam@354 804 | [ H : exec _ _ |- _ ] => inversion H; []; crush
adam@354 805 end.
adam@354 806
adam@354 807 Theorem cleft_identity : forall A B (a : A) (f : A -> comp B),
adam@354 808 comp_eq (Bnd (Ret a) f) (f a).
adam@354 809 red; crush; inverter; eauto.
adam@354 810 Qed.
adam@354 811
adam@354 812 Theorem cright_identity : forall A (m : comp A),
adam@354 813 comp_eq (Bnd m (@Ret _)) m.
adam@354 814 red; crush; inverter; eauto.
adam@354 815 Qed.
adam@354 816
adam@354 817 Lemma cassociativity1 : forall A B C (f : A -> comp B) (g : B -> comp C) r c,
adam@354 818 exec c r
adam@354 819 -> forall m, c = Bnd (Bnd m f) g
adam@354 820 -> exec (Bnd m (fun x => Bnd (f x) g)) r.
adam@354 821 induction 1; crush.
adam@354 822 match goal with
adam@354 823 | [ H : Bnd _ _ = Bnd _ _ |- _ ] => injection H; clear H; intros; try subst
adam@354 824 end.
adam@549 825 try subst B. (* This line expected to fail in Coq 8.4 and succeed in Coq 8.6. *)
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@549 839 try subst A. (* Same as above *)
adam@354 840 crush.
adam@354 841 inversion H0; clear H0; crush.
adam@354 842 eauto.
adam@354 843 Qed.
adam@354 844
adam@354 845 Hint Resolve cassociativity1 cassociativity2.
adam@354 846
adam@354 847 Theorem cassociativity : forall A B C (m : comp A) (f : A -> comp B) (g : B -> comp C),
adam@354 848 comp_eq (Bnd (Bnd m f) g) (Bnd m (fun x => Bnd (f x) g)).
adam@354 849 red; crush; eauto.
adam@354 850 Qed.
adam@356 851 (* end hide *)
adam@356 852
adam@469 853 (** 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@356 854
adam@356 855 Notation "x <- m1 ; m2" := (Bnd m1 (fun x => m2)).
adam@354 856
adam@354 857 CoFixpoint mergeSort'' A (le : A -> A -> bool) (ls : list A) : comp (list A) :=
adam@354 858 if le_lt_dec 2 (length ls)
adam@480 859 then let lss := split ls in
adam@356 860 ls1 <- mergeSort'' le (fst lss);
adam@356 861 ls2 <- mergeSort'' le (snd lss);
adam@356 862 Ret (merge le ls1 ls2)
adam@354 863 else Ret ls.
adam@354 864
adam@356 865 (** To execute this function, we go through the usual exercise of writing a function to catalyze evaluation of co-recursive calls. *)
adam@356 866
adam@354 867 Definition frob' A (c : comp A) :=
adam@354 868 match c with
adam@354 869 | Ret x => Ret x
adam@354 870 | Bnd _ c' f => Bnd c' f
adam@354 871 end.
adam@354 872
adam@354 873 Lemma exec_frob : forall A (c : comp A) x,
adam@354 874 exec (frob' c) x
adam@354 875 -> exec c x.
adam@356 876 destruct c; crush.
adam@354 877 Qed.
adam@354 878
adam@356 879 (** Now the same sort of proof script that we applied for testing [thunk]s will get the job done. *)
adam@356 880
adam@354 881 Lemma test_mergeSort'' : exec (mergeSort'' leb (1 :: 2 :: 36 :: 8 :: 19 :: nil))
adam@354 882 (1 :: 2 :: 8 :: 19 :: 36 :: nil).
adam@354 883 repeat (apply exec_frob; simpl; econstructor).
adam@354 884 Qed.
adam@354 885
adam@356 886 (** 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@356 887
adam@354 888 Definition curriedAdd (n : nat) := Ret (fun m : nat => Ret (n + m)).
adam@354 889
adam@356 890 (** This definition works fine, but we run into trouble when we try to apply it in a trivial way.
adam@356 891 [[
adam@356 892 Definition testCurriedAdd := Bnd (curriedAdd 2) (fun f => f 3).
adam@354 893 ]]
adam@354 894
adam@354 895 <<
adam@354 896 Error: Universe inconsistency.
adam@354 897 >>
adam@356 898
adam@496 899 The problem has to do with rules for inductive definitions that we will 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@354 900
adam@354 901
adam@357 902 (** * Comparing the Alternatives *)
adam@354 903
adam@453 904 (** 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@356 905
adam@356 906 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@356 907
adam@356 908 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@356 909
adam@480 910 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. (For this and other details of notations, see Chapter 12 of the Coq 8.4 manual.)
adam@356 911
adam@465 912 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@356 913
adam@356 914 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@356 915
adam@356 916 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@356 917
adam@354 918 Definition curriedAdd' (n : nat) := Return (fun m : nat => Return (n + m)).
adam@354 919
adam@356 920 Definition testCurriedAdd := Bind (curriedAdd' 2) (fun f => f 3).
adam@356 921
adam@357 922 (** 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@354 923
adam@354 924 Fixpoint map A B (f : A -> computation B) (ls : list A) : computation (list B) :=
adam@354 925 match ls with
adam@354 926 | nil => Return nil
adam@354 927 | x :: ls' => Bind (f x) (fun x' =>
adam@354 928 Bind (map f ls') (fun ls'' =>
adam@354 929 Return (x' :: ls'')))
adam@354 930 end.
adam@354 931
adam@355 932 (** remove printing exists *)
adam@356 933 Theorem test_map : run (map (fun x => Return (S x)) (1 :: 2 :: 3 :: nil))
adam@356 934 (2 :: 3 :: 4 :: nil).
adam@354 935 exists 1; reflexivity.
adam@354 936 Qed.
adam@356 937
adam@524 938 (** One further disadvantage of [comp] is that we cannot prove an inversion lemma for executions of [Bind] without appealing to an _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@356 939
adam@357 940 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. *)