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Note 8.11 port
author Adam Chlipala <adam@chlipala.net>
date Sun, 02 Feb 2020 10:51:18 -0500
parents a913f19955e2
children 1dc1d41620b6
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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@563 15 Require Extraction.
adam@563 16
adam@350 17 Set Implicit Arguments.
adam@534 18 Set Asymmetric Patterns.
adam@350 19 (* end hide *)
adam@350 20
adam@350 21
adam@350 22 (** %\chapter{General Recursion}% *)
adam@350 23
adam@353 24 (** 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 25
adam@424 26 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 27
adam@353 28 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 29
adam@404 30 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 31
adam@351 32
adam@351 33 (** * Well-Founded Recursion *)
adam@351 34
adam@404 35 (** 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 36
adam@351 37 Section mergeSort.
adam@351 38 Variable A : Type.
adam@351 39 Variable le : A -> A -> bool.
adam@475 40
adam@424 41 (** We have a set equipped with some "less-than-or-equal-to" test. *)
adam@351 42
adam@351 43 (** A standard function inserts an element into a sorted list, preserving sortedness. *)
adam@351 44
adam@351 45 Fixpoint insert (x : A) (ls : list A) : list A :=
adam@351 46 match ls with
adam@351 47 | nil => x :: nil
adam@351 48 | h :: ls' =>
adam@351 49 if le x h
adam@351 50 then x :: ls
adam@351 51 else h :: insert x ls'
adam@351 52 end.
adam@351 53
adam@351 54 (** 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 55
adam@351 56 Fixpoint merge (ls1 ls2 : list A) : list A :=
adam@351 57 match ls1 with
adam@351 58 | nil => ls2
adam@351 59 | h :: ls' => insert h (merge ls' ls2)
adam@351 60 end.
adam@351 61
adam@480 62 (** 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 63
adam@480 64 Fixpoint split (ls : list A) : list A * list A :=
adam@351 65 match ls with
adam@351 66 | nil => (nil, nil)
adam@351 67 | h :: nil => (h :: nil, nil)
adam@351 68 | h1 :: h2 :: ls' =>
adam@480 69 let (ls1, ls2) := split ls' in
adam@351 70 (h1 :: ls1, h2 :: ls2)
adam@351 71 end.
adam@351 72
adam@424 73 (** Now, let us try to write the final sorting function, using a natural number "[<=]" test [leb] from the standard library.
adam@351 74 [[
adam@351 75 Fixpoint mergeSort (ls : list A) : list A :=
adam@453 76 if leb (length ls) 1
adam@351 77 then ls
adam@480 78 else let lss := split ls in
adam@351 79 merge (mergeSort (fst lss)) (mergeSort (snd lss)).
adam@351 80 ]]
adam@351 81
adam@351 82 <<
adam@351 83 Recursive call to mergeSort has principal argument equal to
adam@480 84 "fst (split ls)" instead of a subterm of "ls".
adam@351 85 >>
adam@351 86
adam@351 87 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 88
adam@351 89 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 90
adam@351 91 Print well_founded.
adam@351 92 (** %\vspace{-.15in}% [[
adam@351 93 well_founded =
adam@351 94 fun (A : Type) (R : A -> A -> Prop) => forall a : A, Acc R a
adam@351 95 ]]
adam@351 96
adam@404 97 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 98
adam@424 99 (* begin hide *)
adam@437 100 (* begin thide *)
adam@424 101 Definition Acc_intro' := Acc_intro.
adam@437 102 (* end thide *)
adam@424 103 (* end hide *)
adam@424 104
adam@351 105 Print Acc.
adam@351 106 (** %\vspace{-.15in}% [[
adam@351 107 Inductive Acc (A : Type) (R : A -> A -> Prop) (x : A) : Prop :=
adam@351 108 Acc_intro : (forall y : A, R y x -> Acc R y) -> Acc R x
adam@351 109 ]]
adam@351 110
adam@424 111 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@351 112
adam@474 113 CoInductive infiniteDecreasingChain A (R : A -> A -> Prop) : stream A -> Prop :=
adam@474 114 | ChainCons : forall x y s, infiniteDecreasingChain R (Cons y s)
adam@351 115 -> R y x
adam@474 116 -> infiniteDecreasingChain R (Cons x (Cons y s)).
adam@351 117
adam@351 118 (** We can now prove that any accessible element cannot be the beginning of any infinite decreasing chain. *)
adam@351 119
adam@351 120 (* begin thide *)
adam@474 121 Lemma noBadChains' : forall A (R : A -> A -> Prop) x, Acc R x
adam@474 122 -> forall s, ~infiniteDecreasingChain R (Cons x s).
adam@351 123 induction 1; crush;
adam@351 124 match goal with
adam@474 125 | [ H : infiniteDecreasingChain _ _ |- _ ] => inversion H; eauto
adam@351 126 end.
adam@351 127 Qed.
adam@351 128
adam@351 129 (** From here, the absence of infinite decreasing chains in well-founded sets is immediate. *)
adam@351 130
adam@474 131 Theorem noBadChains : forall A (R : A -> A -> Prop), well_founded R
adam@474 132 -> forall s, ~infiniteDecreasingChain R s.
adam@474 133 destruct s; apply noBadChains'; auto.
adam@351 134 Qed.
adam@351 135 (* end thide *)
adam@351 136
adam@351 137 (** 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@351 138
adam@351 139 Check Fix.
adam@351 140 (** %\vspace{-.15in}%[[
adam@351 141 Fix
adam@351 142 : forall (A : Type) (R : A -> A -> Prop),
adam@351 143 well_founded R ->
adam@351 144 forall P : A -> Type,
adam@351 145 (forall x : A, (forall y : A, R y x -> P y) -> P x) ->
adam@351 146 forall x : A, P x
adam@351 147 ]]
adam@351 148
adam@351 149 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@351 150 [[
adam@351 151 forall x : A, (forall y : A, R y x -> P y) -> P x
adam@351 152 ]]
adam@351 153
adam@424 154 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@351 155
adam@353 156 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@351 157
adam@351 158 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@351 159
adam@351 160 Definition lengthOrder (ls1 ls2 : list A) :=
adam@351 161 length ls1 < length ls2.
adam@351 162
adam@353 163 (** 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@351 164
adam@351 165 Hint Constructors Acc.
adam@351 166
adam@351 167 Lemma lengthOrder_wf' : forall len, forall ls, length ls <= len -> Acc lengthOrder ls.
adam@351 168 unfold lengthOrder; induction len; crush.
adam@351 169 Defined.
adam@351 170
adam@351 171 Theorem lengthOrder_wf : well_founded lengthOrder.
adam@351 172 red; intro; eapply lengthOrder_wf'; eauto.
adam@351 173 Defined.
adam@351 174
adam@514 175 (** 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@351 176
adam@480 177 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@351 178
adam@480 179 Lemma split_wf : forall len ls, 2 <= length ls <= len
adam@480 180 -> let (ls1, ls2) := split ls in
adam@351 181 lengthOrder ls1 ls /\ lengthOrder ls2 ls.
adam@351 182 unfold lengthOrder; induction len; crush; do 2 (destruct ls; crush);
adam@351 183 destruct (le_lt_dec 2 (length ls));
adam@351 184 repeat (match goal with
adam@351 185 | [ _ : length ?E < 2 |- _ ] => destruct E
adam@351 186 | [ _ : S (length ?E) < 2 |- _ ] => destruct E
adam@480 187 | [ IH : _ |- context[split ?L] ] =>
adam@480 188 specialize (IH L); destruct (split L); destruct IH
adam@351 189 end; crush).
adam@351 190 Defined.
adam@351 191
adam@480 192 Ltac split_wf := intros ls ?; intros; generalize (@split_wf (length ls) ls);
adam@480 193 destruct (split ls); destruct 1; crush.
adam@351 194
adam@480 195 Lemma split_wf1 : forall ls, 2 <= length ls
adam@480 196 -> lengthOrder (fst (split ls)) ls.
adam@480 197 split_wf.
adam@351 198 Defined.
adam@351 199
adam@480 200 Lemma split_wf2 : forall ls, 2 <= length ls
adam@480 201 -> lengthOrder (snd (split ls)) ls.
adam@480 202 split_wf.
adam@351 203 Defined.
adam@351 204
adam@480 205 Hint Resolve split_wf1 split_wf2.
adam@351 206
adam@453 207 (** 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@351 208
adam@351 209 Definition mergeSort : list A -> list A.
adam@351 210 (* begin thide *)
adam@351 211 refine (Fix lengthOrder_wf (fun _ => list A)
adam@351 212 (fun (ls : list A)
adam@351 213 (mergeSort : forall ls' : list A, lengthOrder ls' ls -> list A) =>
adam@351 214 if le_lt_dec 2 (length ls)
adam@480 215 then let lss := split ls in
adam@351 216 merge (mergeSort (fst lss) _) (mergeSort (snd lss) _)
adam@351 217 else ls)); subst lss; eauto.
adam@351 218 Defined.
adam@351 219 (* end thide *)
adam@351 220 End mergeSort.
adam@351 221
adam@351 222 (** The important thing is that it is now easy to evaluate calls to [mergeSort]. *)
adam@351 223
adam@351 224 Eval compute in mergeSort leb (1 :: 2 :: 36 :: 8 :: 19 :: nil).
adam@351 225 (** [= 1 :: 2 :: 8 :: 19 :: 36 :: nil] *)
adam@351 226
adam@441 227 (** %\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@351 228
adam@351 229 (* begin thide *)
adam@351 230 Theorem mergeSort_eq : forall A (le : A -> A -> bool) ls,
adam@351 231 mergeSort le ls = if le_lt_dec 2 (length ls)
adam@480 232 then let lss := split ls in
adam@351 233 merge le (mergeSort le (fst lss)) (mergeSort le (snd lss))
adam@351 234 else ls.
adam@351 235 intros; apply (Fix_eq (@lengthOrder_wf A) (fun _ => list A)); intros.
adam@351 236
adam@424 237 (** 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@351 238
adam@351 239 Check Fix_eq.
adam@351 240 (** %\vspace{-.15in}%[[
adam@351 241 Fix_eq
adam@351 242 : forall (A : Type) (R : A -> A -> Prop) (Rwf : well_founded R)
adam@351 243 (P : A -> Type)
adam@351 244 (F : forall x : A, (forall y : A, R y x -> P y) -> P x),
adam@351 245 (forall (x : A) (f g : forall y : A, R y x -> P y),
adam@351 246 (forall (y : A) (p : R y x), f y p = g y p) -> F x f = F x g) ->
adam@351 247 forall x : A,
adam@351 248 Fix Rwf P F x = F x (fun (y : A) (_ : R y x) => Fix Rwf P F y)
adam@351 249 ]]
adam@351 250
adam@465 251 Most such obligations are dischargeable with straightforward proof automation, and this example is no exception. *)
adam@351 252
adam@351 253 match goal with
adam@351 254 | [ |- context[match ?E with left _ => _ | right _ => _ end] ] => destruct E
adam@351 255 end; simpl; f_equal; auto.
adam@351 256 Qed.
adam@351 257 (* end thide *)
adam@351 258
adam@351 259 (** As a final test of our definition's suitability, we can extract to OCaml. *)
adam@351 260
adam@351 261 Extraction mergeSort.
adam@351 262
adam@351 263 (** <<
adam@351 264 let rec mergeSort le x =
adam@351 265 match le_lt_dec (S (S O)) (length x) with
adam@351 266 | Left ->
adam@480 267 let lss = split x in
adam@351 268 merge le (mergeSort le (fst lss)) (mergeSort le (snd lss))
adam@351 269 | Right -> x
adam@351 270 >>
adam@351 271
adam@353 272 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@351 273
adam@351 274 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@351 275
adam@351 276 Check well_founded_induction.
adam@351 277 (** %\vspace{-.15in}%[[
adam@351 278 well_founded_induction
adam@351 279 : forall (A : Type) (R : A -> A -> Prop),
adam@351 280 well_founded R ->
adam@351 281 forall P : A -> Set,
adam@351 282 (forall x : A, (forall y : A, R y x -> P y) -> P x) ->
adam@351 283 forall a : A, P a
adam@351 284 ]]
adam@351 285
adam@351 286 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@352 287
adam@352 288
adam@354 289 (** * A Non-Termination Monad Inspired by Domain Theory *)
adam@352 290
adam@424 291 (** 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@355 292
adam@355 293 Consider this definition of a type of computations. *)
adam@355 294
adam@352 295 Section computation.
adam@352 296 Variable A : Type.
adam@355 297 (** The type [A] describes the result a computation will yield, if it terminates.
adam@355 298
adam@355 299 We give a rich dependent type to computations themselves: *)
adam@352 300
adam@352 301 Definition computation :=
adam@352 302 {f : nat -> option A
adam@352 303 | forall (n : nat) (v : A),
adam@352 304 f n = Some v
adam@352 305 -> forall (n' : nat), n' >= n
adam@352 306 -> f n' = Some v}.
adam@352 307
adam@474 308 (** 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@355 309
adam@355 310 It is easy to define a relation characterizing when a computation runs to a particular result at a particular approximation level. *)
adam@355 311
adam@352 312 Definition runTo (m : computation) (n : nat) (v : A) :=
adam@352 313 proj1_sig m n = Some v.
adam@352 314
adam@355 315 (** On top of [runTo], we also define [run], which is the most abstract notion of when a computation runs to a value. *)
adam@355 316
adam@352 317 Definition run (m : computation) (v : A) :=
adam@352 318 exists n, runTo m n v.
adam@352 319 End computation.
adam@352 320
adam@355 321 (** 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@355 322 (* begin hide *)
adam@355 323
adam@352 324 Hint Unfold runTo.
adam@352 325
adam@352 326 Ltac run' := unfold run, runTo in *; try red; crush;
adam@352 327 repeat (match goal with
adam@352 328 | [ _ : proj1_sig ?E _ = _ |- _ ] =>
adam@352 329 match goal with
adam@352 330 | [ x : _ |- _ ] =>
adam@352 331 match x with
adam@352 332 | E => destruct E
adam@352 333 end
adam@352 334 end
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 *; try subst
adam@352 337 | [ _ : context[match ?M with exist _ _ => _ end] |- _ ] => let Heq := fresh "Heq" in
adam@352 338 case_eq M; intros ? ? Heq; try rewrite Heq in *; subst
adam@352 339 | [ H : forall n v, ?E n = Some v -> _,
adam@352 340 _ : context[match ?E ?N with Some _ => _ | None => _ end] |- _ ] =>
adam@426 341 specialize (H N); destruct (E N); try rewrite (H _ (eq_refl _)) by auto; try discriminate
adam@352 342 | [ H : forall n v, ?E n = Some v -> _, H' : ?E _ = Some _ |- _ ] => rewrite (H _ _ H') by auto
adam@352 343 end; simpl in *); eauto 7.
adam@352 344
adam@352 345 Ltac run := run'; repeat (match goal with
adam@352 346 | [ H : forall n v, ?E n = Some v -> _
adam@352 347 |- context[match ?E ?N with Some _ => _ | None => _ end] ] =>
adam@426 348 specialize (H N); destruct (E N); try rewrite (H _ (eq_refl _)) by auto; try discriminate
adam@352 349 end; run').
adam@352 350
adam@352 351 Lemma ex_irrelevant : forall P : Prop, P -> exists n : nat, P.
adam@352 352 exists 0; auto.
adam@352 353 Qed.
adam@352 354
adam@352 355 Hint Resolve ex_irrelevant.
adam@352 356
adam@352 357 Require Import Max.
adam@352 358
adam@380 359 Theorem max_spec_le : forall n m, n <= m /\ max n m = m \/ m <= n /\ max n m = n.
adam@380 360 induction n; destruct m; simpl; intuition;
adam@380 361 specialize (IHn m); intuition.
adam@380 362 Qed.
adam@380 363
adam@352 364 Ltac max := intros n m; generalize (max_spec_le n m); crush.
adam@352 365
adam@352 366 Lemma max_1 : forall n m, max n m >= n.
adam@352 367 max.
adam@352 368 Qed.
adam@352 369
adam@352 370 Lemma max_2 : forall n m, max n m >= m.
adam@352 371 max.
adam@352 372 Qed.
adam@352 373
adam@352 374 Hint Resolve max_1 max_2.
adam@352 375
adam@352 376 Lemma ge_refl : forall n, n >= n.
adam@352 377 crush.
adam@352 378 Qed.
adam@352 379
adam@352 380 Hint Resolve ge_refl.
adam@352 381
adam@352 382 Hint Extern 1 => match goal with
adam@352 383 | [ H : _ = exist _ _ _ |- _ ] => rewrite H
adam@352 384 end.
adam@355 385 (* end hide *)
adam@355 386 (** remove printing exists *)
adam@355 387
adam@480 388 (** 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@352 389
adam@352 390 Section Bottom.
adam@352 391 Variable A : Type.
adam@352 392
adam@352 393 Definition Bottom : computation A.
adam@352 394 exists (fun _ : nat => @None A); abstract run.
adam@352 395 Defined.
adam@352 396
adam@352 397 Theorem run_Bottom : forall v, ~run Bottom v.
adam@352 398 run.
adam@352 399 Qed.
adam@352 400 End Bottom.
adam@352 401
adam@355 402 (** A slightly more complicated example is [Return], which gives the same terminating answer at every approximation level. *)
adam@355 403
adam@352 404 Section Return.
adam@352 405 Variable A : Type.
adam@352 406 Variable v : A.
adam@352 407
adam@352 408 Definition Return : computation A.
adam@352 409 intros; exists (fun _ : nat => Some v); abstract run.
adam@352 410 Defined.
adam@352 411
adam@352 412 Theorem run_Return : run Return v.
adam@352 413 run.
adam@352 414 Qed.
adam@352 415 End Return.
adam@352 416
adam@474 417 (** 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@352 418
adam@352 419 Section Bind.
adam@352 420 Variables A B : Type.
adam@352 421 Variable m1 : computation A.
adam@352 422 Variable m2 : A -> computation B.
adam@352 423
adam@352 424 Definition Bind : computation B.
adam@352 425 exists (fun n =>
adam@357 426 let (f1, _) := m1 in
adam@352 427 match f1 n with
adam@352 428 | None => None
adam@352 429 | Some v =>
adam@357 430 let (f2, _) := m2 v in
adam@352 431 f2 n
adam@352 432 end); abstract run.
adam@352 433 Defined.
adam@352 434
adam@352 435 Theorem run_Bind : forall (v1 : A) (v2 : B),
adam@352 436 run m1 v1
adam@352 437 -> run (m2 v1) v2
adam@352 438 -> run Bind v2.
adam@352 439 run; match goal with
adam@352 440 | [ x : nat, y : nat |- _ ] => exists (max x y)
adam@352 441 end; run.
adam@352 442 Qed.
adam@352 443 End Bind.
adam@352 444
adam@355 445 (** A simple notation lets us write [Bind] calls the way they appear in Haskell. *)
adam@352 446
adam@352 447 Notation "x <- m1 ; m2" :=
adam@352 448 (Bind m1 (fun x => m2)) (right associativity, at level 70).
adam@352 449
adam@424 450 (** 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@355 451
adam@352 452 Definition meq A (m1 m2 : computation A) := forall n, proj1_sig m1 n = proj1_sig m2 n.
adam@352 453
adam@352 454 Theorem left_identity : forall A B (a : A) (f : A -> computation B),
adam@352 455 meq (Bind (Return a) f) (f a).
adam@352 456 run.
adam@352 457 Qed.
adam@352 458
adam@352 459 Theorem right_identity : forall A (m : computation A),
adam@352 460 meq (Bind m (@Return _)) m.
adam@352 461 run.
adam@352 462 Qed.
adam@352 463
adam@357 464 Theorem associativity : forall A B C (m : computation A)
adam@357 465 (f : A -> computation B) (g : B -> computation C),
adam@352 466 meq (Bind (Bind m f) g) (Bind m (fun x => Bind (f x) g)).
adam@352 467 run.
adam@352 468 Qed.
adam@352 469
adam@398 470 (** 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@352 471
adam@352 472 Section lattice.
adam@352 473 Variable A : Type.
adam@352 474
adam@352 475 Definition leq (x y : option A) :=
adam@352 476 forall v, x = Some v -> y = Some v.
adam@352 477 End lattice.
adam@352 478
adam@355 479 (** 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@352 480
adam@352 481 Section Fix.
adam@475 482
adam@355 483 (** First, we have the function domain and range types. *)
adam@355 484
adam@352 485 Variables A B : Type.
adam@355 486
adam@355 487 (** Next comes the function body, which is written as though it can be parameterized over itself, for recursive calls. *)
adam@355 488
adam@352 489 Variable f : (A -> computation B) -> (A -> computation B).
adam@352 490
adam@355 491 (** 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@355 492
adam@352 493 Hypothesis f_continuous : forall n v v1 x,
adam@352 494 runTo (f v1 x) n v
adam@352 495 -> forall (v2 : A -> computation B),
adam@352 496 (forall x, leq (proj1_sig (v1 x) n) (proj1_sig (v2 x) n))
adam@352 497 -> runTo (f v2 x) n v.
adam@352 498
adam@355 499 (** 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@355 500
adam@352 501 Fixpoint Fix' (n : nat) (x : A) : computation B :=
adam@352 502 match n with
adam@352 503 | O => Bottom _
adam@352 504 | S n' => f (Fix' n') x
adam@352 505 end.
adam@352 506
adam@355 507 (** Now it is straightforward to package [Fix'] as a computation combinator [Fix]. *)
adam@355 508
adam@352 509 Hint Extern 1 (_ >= _) => omega.
adam@352 510 Hint Unfold leq.
adam@352 511
adam@352 512 Lemma Fix'_ok : forall steps n x v, proj1_sig (Fix' n x) steps = Some v
adam@352 513 -> forall n', n' >= n
adam@352 514 -> proj1_sig (Fix' n' x) steps = Some v.
adam@352 515 unfold runTo in *; induction n; crush;
adam@352 516 match goal with
adam@352 517 | [ H : _ >= _ |- _ ] => inversion H; crush; eauto
adam@352 518 end.
adam@352 519 Qed.
adam@352 520
adam@352 521 Hint Resolve Fix'_ok.
adam@352 522
adam@352 523 Hint Extern 1 (proj1_sig _ _ = _) => simpl;
adam@352 524 match goal with
adam@352 525 | [ |- proj1_sig ?E _ = _ ] => eapply (proj2_sig E)
adam@352 526 end.
adam@352 527
adam@352 528 Definition Fix : A -> computation B.
adam@352 529 intro x; exists (fun n => proj1_sig (Fix' n x) n); abstract run.
adam@352 530 Defined.
adam@352 531
adam@355 532 (** Finally, we can prove that [Fix] obeys the expected computation rule. *)
adam@352 533
adam@352 534 Theorem run_Fix : forall x v,
adam@352 535 run (f Fix x) v
adam@352 536 -> run (Fix x) v.
adam@352 537 run; match goal with
adam@352 538 | [ n : nat |- _ ] => exists (S n); eauto
adam@352 539 end.
adam@352 540 Qed.
adam@352 541 End Fix.
adam@352 542
adam@355 543 (* begin hide *)
adam@352 544 Lemma leq_Some : forall A (x y : A), leq (Some x) (Some y)
adam@352 545 -> x = y.
adam@426 546 intros ? ? ? H; generalize (H _ (eq_refl _)); crush.
adam@352 547 Qed.
adam@352 548
adam@352 549 Lemma leq_None : forall A (x y : A), leq (Some x) None
adam@352 550 -> False.
adam@426 551 intros ? ? ? H; generalize (H _ (eq_refl _)); crush.
adam@352 552 Qed.
adam@352 553
adam@355 554 Ltac mergeSort' := run;
adam@355 555 repeat (match goal with
adam@355 556 | [ |- context[match ?E with O => _ | S _ => _ end] ] => destruct E
adam@355 557 end; run);
adam@355 558 repeat match goal with
adam@355 559 | [ H : forall x, leq (proj1_sig (?f x) _) (proj1_sig (?g x) _) |- _ ] =>
adam@355 560 match goal with
adam@355 561 | [ H1 : f ?arg = _, H2 : g ?arg = _ |- _ ] =>
adam@355 562 generalize (H arg); rewrite H1; rewrite H2; clear H1 H2; simpl; intro
adam@355 563 end
adam@355 564 end; run; repeat match goal with
adam@355 565 | [ H : _ |- _ ] => (apply leq_None in H; tauto) || (apply leq_Some in H; subst)
adam@355 566 end; auto.
adam@355 567 (* end hide *)
adam@355 568
adam@355 569 (** 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@355 570
adam@352 571 Definition mergeSort' : forall A, (A -> A -> bool) -> list A -> computation (list A).
adam@352 572 refine (fun A le => Fix
adam@352 573 (fun (mergeSort : list A -> computation (list A))
adam@352 574 (ls : list A) =>
adam@352 575 if le_lt_dec 2 (length ls)
adam@480 576 then let lss := split ls in
adam@352 577 ls1 <- mergeSort (fst lss);
adam@352 578 ls2 <- mergeSort (snd lss);
adam@352 579 Return (merge le ls1 ls2)
adam@355 580 else Return ls) _); abstract mergeSort'.
adam@352 581 Defined.
adam@352 582
adam@424 583 (** 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@352 584
adam@352 585 Lemma test_mergeSort' : run (mergeSort' leb (1 :: 2 :: 36 :: 8 :: 19 :: nil))
adam@352 586 (1 :: 2 :: 8 :: 19 :: 36 :: nil).
adam@352 587 exists 4; reflexivity.
adam@352 588 Qed.
adam@352 589
adam@453 590 (** 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@355 591
adam@355 592 (* begin hide *)
adam@355 593 Ltac looper := unfold leq in *; run;
adam@355 594 repeat match goal with
adam@355 595 | [ x : unit |- _ ] => destruct x
adam@355 596 | [ x : bool |- _ ] => destruct x
adam@355 597 end; auto.
adam@355 598 (* end hide *)
adam@355 599
adam@352 600 Definition looper : bool -> computation unit.
adam@352 601 refine (Fix (fun looper (b : bool) =>
adam@355 602 if b then Return tt else looper b) _); abstract looper.
adam@352 603 Defined.
adam@352 604
adam@352 605 Lemma test_looper : run (looper true) tt.
adam@352 606 exists 1; reflexivity.
adam@352 607 Qed.
adam@354 608
adam@355 609 (** As before, proving outputs for specific inputs is as easy as demonstrating a high enough approximation level.
adam@355 610
adam@480 611 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@355 612
adam@354 613
adam@354 614 (** * Co-Inductive Non-Termination Monads *)
adam@354 615
adam@356 616 (** 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@356 617
adam@356 618 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@356 619
adam@354 620 CoInductive thunk (A : Type) : Type :=
adam@354 621 | Answer : A -> thunk A
adam@354 622 | Think : thunk A -> thunk A.
adam@354 623
adam@356 624 (** 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@356 625
adam@424 626 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@356 627
adam@356 628 CoFixpoint TBind A B (m1 : thunk A) (m2 : A -> thunk B) : thunk B :=
adam@354 629 match m1 with
adam@354 630 | Answer x => m2 x
adam@354 631 | Think m1' => Think (TBind m1' m2)
adam@354 632 end.
adam@354 633
adam@356 634 (** 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@356 635
adam@356 636 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@356 637
adam@356 638 (* begin hide *)
adam@354 639 CoInductive thunk_eq A : thunk A -> thunk A -> Prop :=
adam@354 640 | EqAnswer : forall x, thunk_eq (Answer x) (Answer x)
adam@354 641 | EqThinkL : forall m1 m2, thunk_eq m1 m2 -> thunk_eq (Think m1) m2
adam@354 642 | EqThinkR : forall m1 m2, thunk_eq m1 m2 -> thunk_eq m1 (Think m2).
adam@354 643
adam@354 644 Section thunk_eq_coind.
adam@354 645 Variable A : Type.
adam@354 646 Variable P : thunk A -> thunk A -> Prop.
adam@354 647
adam@354 648 Hypothesis H : forall m1 m2, P m1 m2
adam@354 649 -> match m1, m2 with
adam@354 650 | Answer x1, Answer x2 => x1 = x2
adam@354 651 | Think m1', Think m2' => P m1' m2'
adam@354 652 | Think m1', _ => P m1' m2
adam@354 653 | _, Think m2' => P m1 m2'
adam@354 654 end.
adam@354 655
adam@354 656 Theorem thunk_eq_coind : forall m1 m2, P m1 m2 -> thunk_eq m1 m2.
adam@568 657 cofix thunk_eq_coind; intros;
adam@354 658 match goal with
adam@354 659 | [ H' : P _ _ |- _ ] => specialize (H H'); clear H'
adam@354 660 end; destruct m1; destruct m2; subst; repeat constructor; auto.
adam@354 661 Qed.
adam@354 662 End thunk_eq_coind.
adam@356 663 (* end hide *)
adam@356 664
adam@356 665 (** 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@354 666
adam@354 667 Definition frob A (m : thunk A) : thunk A :=
adam@354 668 match m with
adam@354 669 | Answer x => Answer x
adam@354 670 | Think m' => Think m'
adam@354 671 end.
adam@354 672
adam@354 673 Theorem frob_eq : forall A (m : thunk A), frob m = m.
adam@354 674 destruct m; reflexivity.
adam@354 675 Qed.
adam@354 676
adam@356 677 (* begin hide *)
adam@354 678 Theorem thunk_eq_frob : forall A (m1 m2 : thunk A),
adam@354 679 thunk_eq (frob m1) (frob m2)
adam@354 680 -> thunk_eq m1 m2.
adam@354 681 intros; repeat rewrite frob_eq in *; auto.
adam@354 682 Qed.
adam@354 683
adam@354 684 Ltac findDestr := match goal with
adam@354 685 | [ |- context[match ?E with Answer _ => _ | Think _ => _ end] ] =>
adam@354 686 match E with
adam@354 687 | context[match _ with Answer _ => _ | Think _ => _ end] => fail 1
adam@354 688 | _ => destruct E
adam@354 689 end
adam@354 690 end.
adam@354 691
adam@354 692 Theorem thunk_eq_refl : forall A (m : thunk A), thunk_eq m m.
adam@354 693 intros; apply (thunk_eq_coind (fun m1 m2 => m1 = m2)); crush; findDestr; reflexivity.
adam@354 694 Qed.
adam@354 695
adam@354 696 Hint Resolve thunk_eq_refl.
adam@354 697
adam@354 698 Theorem tleft_identity : forall A B (a : A) (f : A -> thunk B),
adam@354 699 thunk_eq (TBind (Answer a) f) (f a).
adam@354 700 intros; apply thunk_eq_frob; crush.
adam@354 701 Qed.
adam@354 702
adam@354 703 Theorem tright_identity : forall A (m : thunk A),
adam@354 704 thunk_eq (TBind m (@Answer _)) m.
adam@354 705 intros; apply (thunk_eq_coind (fun m1 m2 => m1 = TBind m2 (@Answer _))); crush;
adam@354 706 findDestr; reflexivity.
adam@354 707 Qed.
adam@354 708
adam@354 709 Lemma TBind_Answer : forall (A B : Type) (v : A) (m2 : A -> thunk B),
adam@354 710 TBind (Answer v) m2 = m2 v.
adam@354 711 intros; rewrite <- (frob_eq (TBind (Answer v) m2));
adam@354 712 simpl; findDestr; reflexivity.
adam@354 713 Qed.
adam@354 714
adam@375 715 Hint Rewrite TBind_Answer.
adam@354 716
adam@355 717 (** printing exists $\exists$ *)
adam@355 718
adam@354 719 Theorem tassociativity : forall A B C (m : thunk A) (f : A -> thunk B) (g : B -> thunk C),
adam@354 720 thunk_eq (TBind (TBind m f) g) (TBind m (fun x => TBind (f x) g)).
adam@354 721 intros; apply (thunk_eq_coind (fun m1 m2 => (exists m,
adam@354 722 m1 = TBind (TBind m f) g
adam@354 723 /\ m2 = TBind m (fun x => TBind (f x) g))
adam@354 724 \/ m1 = m2)); crush; eauto; repeat (findDestr; crush; eauto).
adam@354 725 Qed.
adam@356 726 (* end hide *)
adam@356 727
adam@356 728 (** As a simple example, here is how we might define a tail-recursive factorial function. *)
adam@354 729
adam@354 730 CoFixpoint fact (n acc : nat) : thunk nat :=
adam@354 731 match n with
adam@354 732 | O => Answer acc
adam@354 733 | S n' => Think (fact n' (S n' * acc))
adam@354 734 end.
adam@354 735
adam@356 736 (** To test our definition, we need an evaluation relation that characterizes results of evaluating [thunk]s. *)
adam@356 737
adam@354 738 Inductive eval A : thunk A -> A -> Prop :=
adam@354 739 | EvalAnswer : forall x, eval (Answer x) x
adam@354 740 | EvalThink : forall m x, eval m x -> eval (Think m) x.
adam@354 741
adam@375 742 Hint Rewrite frob_eq.
adam@354 743
adam@354 744 Lemma eval_frob : forall A (c : thunk A) x,
adam@354 745 eval (frob c) x
adam@354 746 -> eval c x.
adam@354 747 crush.
adam@354 748 Qed.
adam@354 749
adam@354 750 Theorem eval_fact : eval (fact 5 1) 120.
adam@354 751 repeat (apply eval_frob; simpl; constructor).
adam@354 752 Qed.
adam@354 753
adam@356 754 (** We need to apply constructors of [eval] explicitly, but the process is easy to automate completely for concrete input programs.
adam@356 755
adam@465 756 Now consider another very similar definition, this time of a Fibonacci number function. *)
adam@357 757
adam@357 758 Notation "x <- m1 ; m2" :=
adam@357 759 (TBind m1 (fun x => m2)) (right associativity, at level 70).
adam@357 760
adam@404 761 (* begin hide *)
adam@437 762 (* begin thide *)
adam@424 763 Definition fib := pred.
adam@437 764 (* end thide *)
adam@404 765 (* end hide *)
adam@404 766
adam@453 767 (** %\vspace{-.3in}%[[
adam@354 768 CoFixpoint fib (n : nat) : thunk nat :=
adam@354 769 match n with
adam@354 770 | 0 => Answer 1
adam@354 771 | 1 => Answer 1
adam@357 772 | _ => n1 <- fib (pred n);
adam@357 773 n2 <- fib (pred (pred n));
adam@357 774 Answer (n1 + n2)
adam@354 775 end.
adam@354 776 ]]
adam@354 777
adam@356 778 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@356 779
adam@356 780 %\medskip%
adam@356 781
adam@424 782 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@354 783
adam@572 784 Set Universe Polymorphism.
adam@572 785
adam@354 786 CoInductive comp (A : Type) : Type :=
adam@354 787 | Ret : A -> comp A
adam@354 788 | Bnd : forall B, comp B -> (B -> comp A) -> comp A.
adam@354 789
adam@404 790 (** 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@356 791
adam@356 792 It is easy to define the semantics of terminating [comp] computations. *)
adam@356 793
adam@354 794 Inductive exec A : comp A -> A -> Prop :=
adam@354 795 | ExecRet : forall x, exec (Ret x) x
adam@354 796 | ExecBnd : forall B (c : comp B) (f : B -> comp A) x1 x2, exec (A := B) c x1
adam@354 797 -> exec (f x1) x2
adam@354 798 -> exec (Bnd c f) x2.
adam@354 799
adam@356 800 (** 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@356 801
adam@356 802 (* begin hide *)
adam@354 803 Hint Constructors exec.
adam@354 804
adam@354 805 Definition comp_eq A (c1 c2 : comp A) := forall r, exec c1 r <-> exec c2 r.
adam@354 806
adam@354 807 Ltac inverter := repeat match goal with
adam@354 808 | [ H : exec _ _ |- _ ] => inversion H; []; crush
adam@354 809 end.
adam@354 810
adam@354 811 Theorem cleft_identity : forall A B (a : A) (f : A -> comp B),
adam@354 812 comp_eq (Bnd (Ret a) f) (f a).
adam@354 813 red; crush; inverter; eauto.
adam@354 814 Qed.
adam@354 815
adam@354 816 Theorem cright_identity : forall A (m : comp A),
adam@354 817 comp_eq (Bnd m (@Ret _)) m.
adam@354 818 red; crush; inverter; eauto.
adam@354 819 Qed.
adam@354 820
adam@354 821 Lemma cassociativity1 : forall A B C (f : A -> comp B) (g : B -> comp C) r c,
adam@354 822 exec c r
adam@354 823 -> forall m, c = Bnd (Bnd m f) g
adam@354 824 -> exec (Bnd m (fun x => Bnd (f x) g)) r.
adam@354 825 induction 1; crush.
adam@354 826 match goal with
adam@354 827 | [ H : Bnd _ _ = Bnd _ _ |- _ ] => injection H; clear H; intros; try subst
adam@354 828 end.
adam@549 829 try subst B. (* This line expected to fail in Coq 8.4 and succeed in Coq 8.6. *)
adam@354 830 crush.
adam@354 831 inversion H; clear H; crush.
adam@354 832 eauto.
adam@354 833 Qed.
adam@354 834
adam@354 835 Lemma cassociativity2 : forall A B C (f : A -> comp B) (g : B -> comp C) r c,
adam@354 836 exec c r
adam@354 837 -> forall m, c = Bnd m (fun x => Bnd (f x) g)
adam@354 838 -> exec (Bnd (Bnd m f) g) r.
adam@354 839 induction 1; crush.
adam@354 840 match goal with
adam@354 841 | [ H : Bnd _ _ = Bnd _ _ |- _ ] => injection H; clear H; intros; try subst
adam@354 842 end.
adam@549 843 try subst A. (* Same as above *)
adam@354 844 crush.
adam@354 845 inversion H0; clear H0; crush.
adam@354 846 eauto.
adam@354 847 Qed.
adam@354 848
adam@354 849 Hint Resolve cassociativity1 cassociativity2.
adam@354 850
adam@354 851 Theorem cassociativity : forall A B C (m : comp A) (f : A -> comp B) (g : B -> comp C),
adam@354 852 comp_eq (Bnd (Bnd m f) g) (Bnd m (fun x => Bnd (f x) g)).
adam@354 853 red; crush; eauto.
adam@354 854 Qed.
adam@356 855 (* end hide *)
adam@356 856
adam@469 857 (** 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@356 858
adam@356 859 Notation "x <- m1 ; m2" := (Bnd m1 (fun x => m2)).
adam@354 860
adam@354 861 CoFixpoint mergeSort'' A (le : A -> A -> bool) (ls : list A) : comp (list A) :=
adam@354 862 if le_lt_dec 2 (length ls)
adam@480 863 then let lss := split ls in
adam@356 864 ls1 <- mergeSort'' le (fst lss);
adam@356 865 ls2 <- mergeSort'' le (snd lss);
adam@356 866 Ret (merge le ls1 ls2)
adam@354 867 else Ret ls.
adam@354 868
adam@356 869 (** To execute this function, we go through the usual exercise of writing a function to catalyze evaluation of co-recursive calls. *)
adam@356 870
adam@354 871 Definition frob' A (c : comp A) :=
adam@354 872 match c with
adam@354 873 | Ret x => Ret x
adam@354 874 | Bnd _ c' f => Bnd c' f
adam@354 875 end.
adam@354 876
adam@354 877 Lemma exec_frob : forall A (c : comp A) x,
adam@354 878 exec (frob' c) x
adam@354 879 -> exec c x.
adam@356 880 destruct c; crush.
adam@354 881 Qed.
adam@354 882
adam@356 883 (** Now the same sort of proof script that we applied for testing [thunk]s will get the job done. *)
adam@356 884
adam@354 885 Lemma test_mergeSort'' : exec (mergeSort'' leb (1 :: 2 :: 36 :: 8 :: 19 :: nil))
adam@354 886 (1 :: 2 :: 8 :: 19 :: 36 :: nil).
adam@354 887 repeat (apply exec_frob; simpl; econstructor).
adam@354 888 Qed.
adam@354 889
adam@356 890 (** 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@356 891
adam@354 892 Definition curriedAdd (n : nat) := Ret (fun m : nat => Ret (n + m)).
adam@354 893
adam@356 894 (** This definition works fine, but we run into trouble when we try to apply it in a trivial way.
adam@356 895 [[
adam@356 896 Definition testCurriedAdd := Bnd (curriedAdd 2) (fun f => f 3).
adam@354 897 ]]
adam@354 898
adam@354 899 <<
adam@354 900 Error: Universe inconsistency.
adam@354 901 >>
adam@356 902
adam@496 903 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@354 904
adam@354 905
adam@357 906 (** * Comparing the Alternatives *)
adam@354 907
adam@453 908 (** 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@356 909
adam@356 910 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@356 911
adam@356 912 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@356 913
adam@480 914 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@356 915
adam@465 916 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@356 917
adam@356 918 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@356 919
adam@356 920 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@356 921
adam@354 922 Definition curriedAdd' (n : nat) := Return (fun m : nat => Return (n + m)).
adam@354 923
adam@356 924 Definition testCurriedAdd := Bind (curriedAdd' 2) (fun f => f 3).
adam@356 925
adam@357 926 (** 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@354 927
adam@354 928 Fixpoint map A B (f : A -> computation B) (ls : list A) : computation (list B) :=
adam@354 929 match ls with
adam@354 930 | nil => Return nil
adam@354 931 | x :: ls' => Bind (f x) (fun x' =>
adam@354 932 Bind (map f ls') (fun ls'' =>
adam@354 933 Return (x' :: ls'')))
adam@354 934 end.
adam@354 935
adam@355 936 (** remove printing exists *)
adam@356 937 Theorem test_map : run (map (fun x => Return (S x)) (1 :: 2 :: 3 :: nil))
adam@356 938 (2 :: 3 :: 4 :: nil).
adam@354 939 exists 1; reflexivity.
adam@354 940 Qed.
adam@356 941
adam@524 942 (** 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@356 943
adam@357 944 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. *)