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author | Adam Chlipala <adam@chlipala.net> |
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date | Sun, 02 Feb 2014 12:48:15 -0500 |

parents | 3b21f4395178 |

children | ed829eaa91b2 |

rev | line source |
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adam@380 | 1 (* Copyright (c) 2006, 2011-2012, Adam Chlipala |

adam@350 | 2 * |

adam@350 | 3 * This work is licensed under a |

adam@350 | 4 * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 |

adam@350 | 5 * Unported License. |

adam@350 | 6 * The license text is available at: |

adam@350 | 7 * http://creativecommons.org/licenses/by-nc-nd/3.0/ |

adam@350 | 8 *) |

adam@350 | 9 |

adam@350 | 10 (* begin hide *) |

adam@351 | 11 Require Import Arith List. |

adam@350 | 12 |

adam@351 | 13 Require Import CpdtTactics Coinductive. |

adam@350 | 14 |

adam@350 | 15 Set Implicit Arguments. |

adam@350 | 16 (* end hide *) |

adam@350 | 17 |

adam@350 | 18 |

adam@350 | 19 (** %\chapter{General Recursion}% *) |

adam@350 | 20 |

adam@353 | 21 (** Termination of all programs is a crucial property of Gallina. Non-terminating programs introduce logical inconsistency, where any theorem can be proved with an infinite loop. Coq uses a small set of conservative, syntactic criteria to check termination of all recursive definitions. These criteria are insufficient to support the natural encodings of a variety of important programming idioms. Further, since Coq makes it so convenient to encode mathematics computationally, with functional programs, we may find ourselves wanting to employ more complicated recursion in mathematical definitions. |

adam@351 | 22 |

adam@424 | 23 What exactly are the conservative criteria that we run up against? For _recursive_ definitions, recursive calls are only allowed on _syntactic subterms_ of the original primary argument, a restriction known as%\index{primitive recursion}% _primitive recursion_. In fact, Coq's handling of reflexive inductive types (those defined in terms of functions returning the same type) gives a bit more flexibility than in traditional primitive recursion, but the term is still applied commonly. In Chapter 5, we saw how _co-recursive_ definitions are checked against a syntactic guardedness condition that guarantees productivity. |

adam@351 | 24 |

adam@353 | 25 Many natural recursion patterns satisfy neither condition. For instance, there is our simple running example in this chapter, merge sort. We will study three different approaches to more flexible recursion, and the latter two of the approaches will even support definitions that may fail to terminate on certain inputs, without any up-front characterization of which inputs those may be. |

adam@351 | 26 |

adam@404 | 27 Before proceeding, it is important to note that the problem here is not as fundamental as it may appear. The final example of Chapter 5 demonstrated what is called a%\index{deep embedding}% _deep embedding_ of the syntax and semantics of a programming language. That is, we gave a mathematical definition of a language of programs and their meanings. This language clearly admitted non-termination, and we could think of writing all our sophisticated recursive functions with such explicit syntax types. However, in doing so, we forfeit our chance to take advantage of Coq's very good built-in support for reasoning about Gallina programs. We would rather use a%\index{shallow embedding}% _shallow embedding_, where we model informal constructs by encoding them as normal Gallina programs. Each of the three techniques of this chapter follows that style. *) |

adam@351 | 28 |

adam@351 | 29 |

adam@351 | 30 (** * Well-Founded Recursion *) |

adam@351 | 31 |

adam@404 | 32 (** The essence of terminating recursion is that there are no infinite chains of nested recursive calls. This intuition is commonly mapped to the mathematical idea of a%\index{well-founded relation}% _well-founded relation_, and the associated standard technique in Coq is%\index{well-founded recursion}% _well-founded recursion_. The syntactic-subterm relation that Coq applies by default is well-founded, but many cases demand alternate well-founded relations. To demonstrate, let us see where we get stuck on attempting a standard merge sort implementation. *) |

adam@351 | 33 |

adam@351 | 34 Section mergeSort. |

adam@351 | 35 Variable A : Type. |

adam@351 | 36 Variable le : A -> A -> bool. |

adam@475 | 37 |

adam@424 | 38 (** We have a set equipped with some "less-than-or-equal-to" test. *) |

adam@351 | 39 |

adam@351 | 40 (** A standard function inserts an element into a sorted list, preserving sortedness. *) |

adam@351 | 41 |

adam@351 | 42 Fixpoint insert (x : A) (ls : list A) : list A := |

adam@351 | 43 match ls with |

adam@351 | 44 | nil => x :: nil |

adam@351 | 45 | h :: ls' => |

adam@351 | 46 if le x h |

adam@351 | 47 then x :: ls |

adam@351 | 48 else h :: insert x ls' |

adam@351 | 49 end. |

adam@351 | 50 |

adam@351 | 51 (** 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 | 52 |

adam@351 | 53 Fixpoint merge (ls1 ls2 : list A) : list A := |

adam@351 | 54 match ls1 with |

adam@351 | 55 | nil => ls2 |

adam@351 | 56 | h :: ls' => insert h (merge ls' ls2) |

adam@351 | 57 end. |

adam@351 | 58 |

adam@480 | 59 (** 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 | 60 |

adam@480 | 61 Fixpoint split (ls : list A) : list A * list A := |

adam@351 | 62 match ls with |

adam@351 | 63 | nil => (nil, nil) |

adam@351 | 64 | h :: nil => (h :: nil, nil) |

adam@351 | 65 | h1 :: h2 :: ls' => |

adam@480 | 66 let (ls1, ls2) := split ls' in |

adam@351 | 67 (h1 :: ls1, h2 :: ls2) |

adam@351 | 68 end. |

adam@351 | 69 |

adam@424 | 70 (** Now, let us try to write the final sorting function, using a natural number "[<=]" test [leb] from the standard library. |

adam@351 | 71 [[ |

adam@351 | 72 Fixpoint mergeSort (ls : list A) : list A := |

adam@453 | 73 if leb (length ls) 1 |

adam@351 | 74 then ls |

adam@480 | 75 else let lss := split ls in |

adam@351 | 76 merge (mergeSort (fst lss)) (mergeSort (snd lss)). |

adam@351 | 77 ]] |

adam@351 | 78 |

adam@351 | 79 << |

adam@351 | 80 Recursive call to mergeSort has principal argument equal to |

adam@480 | 81 "fst (split ls)" instead of a subterm of "ls". |

adam@351 | 82 >> |

adam@351 | 83 |

adam@351 | 84 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 | 85 |

adam@351 | 86 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 | 87 |

adam@351 | 88 Print well_founded. |

adam@351 | 89 (** %\vspace{-.15in}% [[ |

adam@351 | 90 well_founded = |

adam@351 | 91 fun (A : Type) (R : A -> A -> Prop) => forall a : A, Acc R a |

adam@351 | 92 ]] |

adam@351 | 93 |

adam@404 | 94 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 | 95 |

adam@424 | 96 (* begin hide *) |

adam@437 | 97 (* begin thide *) |

adam@424 | 98 Definition Acc_intro' := Acc_intro. |

adam@437 | 99 (* end thide *) |

adam@424 | 100 (* end hide *) |

adam@424 | 101 |

adam@351 | 102 Print Acc. |

adam@351 | 103 (** %\vspace{-.15in}% [[ |

adam@351 | 104 Inductive Acc (A : Type) (R : A -> A -> Prop) (x : A) : Prop := |

adam@351 | 105 Acc_intro : (forall y : A, R y x -> Acc R y) -> Acc R x |

adam@351 | 106 ]] |

adam@351 | 107 |

adam@424 | 108 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 | 109 |

adam@474 | 110 CoInductive infiniteDecreasingChain A (R : A -> A -> Prop) : stream A -> Prop := |

adam@474 | 111 | ChainCons : forall x y s, infiniteDecreasingChain R (Cons y s) |

adam@351 | 112 -> R y x |

adam@474 | 113 -> infiniteDecreasingChain R (Cons x (Cons y s)). |

adam@351 | 114 |

adam@351 | 115 (** We can now prove that any accessible element cannot be the beginning of any infinite decreasing chain. *) |

adam@351 | 116 |

adam@351 | 117 (* begin thide *) |

adam@474 | 118 Lemma noBadChains' : forall A (R : A -> A -> Prop) x, Acc R x |

adam@474 | 119 -> forall s, ~infiniteDecreasingChain R (Cons x s). |

adam@351 | 120 induction 1; crush; |

adam@351 | 121 match goal with |

adam@474 | 122 | [ H : infiniteDecreasingChain _ _ |- _ ] => inversion H; eauto |

adam@351 | 123 end. |

adam@351 | 124 Qed. |

adam@351 | 125 |

adam@351 | 126 (** From here, the absence of infinite decreasing chains in well-founded sets is immediate. *) |

adam@351 | 127 |

adam@474 | 128 Theorem noBadChains : forall A (R : A -> A -> Prop), well_founded R |

adam@474 | 129 -> forall s, ~infiniteDecreasingChain R s. |

adam@474 | 130 destruct s; apply noBadChains'; auto. |

adam@351 | 131 Qed. |

adam@351 | 132 (* end thide *) |

adam@351 | 133 |

adam@351 | 134 (** 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 | 135 |

adam@351 | 136 Check Fix. |

adam@351 | 137 (** %\vspace{-.15in}%[[ |

adam@351 | 138 Fix |

adam@351 | 139 : forall (A : Type) (R : A -> A -> Prop), |

adam@351 | 140 well_founded R -> |

adam@351 | 141 forall P : A -> Type, |

adam@351 | 142 (forall x : A, (forall y : A, R y x -> P y) -> P x) -> |

adam@351 | 143 forall x : A, P x |

adam@351 | 144 ]] |

adam@351 | 145 |

adam@351 | 146 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 | 147 [[ |

adam@351 | 148 forall x : A, (forall y : A, R y x -> P y) -> P x |

adam@351 | 149 ]] |

adam@351 | 150 |

adam@424 | 151 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 | 152 |

adam@353 | 153 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 | 154 |

adam@351 | 155 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 | 156 |

adam@351 | 157 Definition lengthOrder (ls1 ls2 : list A) := |

adam@351 | 158 length ls1 < length ls2. |

adam@351 | 159 |

adam@353 | 160 (** 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 | 161 |

adam@351 | 162 Hint Constructors Acc. |

adam@351 | 163 |

adam@351 | 164 Lemma lengthOrder_wf' : forall len, forall ls, length ls <= len -> Acc lengthOrder ls. |

adam@351 | 165 unfold lengthOrder; induction len; crush. |

adam@351 | 166 Defined. |

adam@351 | 167 |

adam@351 | 168 Theorem lengthOrder_wf : well_founded lengthOrder. |

adam@351 | 169 red; intro; eapply lengthOrder_wf'; eauto. |

adam@351 | 170 Defined. |

adam@351 | 171 |

adam@514 | 172 (** 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 | 173 |

adam@480 | 174 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 | 175 |

adam@480 | 176 Lemma split_wf : forall len ls, 2 <= length ls <= len |

adam@480 | 177 -> let (ls1, ls2) := split ls in |

adam@351 | 178 lengthOrder ls1 ls /\ lengthOrder ls2 ls. |

adam@351 | 179 unfold lengthOrder; induction len; crush; do 2 (destruct ls; crush); |

adam@351 | 180 destruct (le_lt_dec 2 (length ls)); |

adam@351 | 181 repeat (match goal with |

adam@351 | 182 | [ _ : length ?E < 2 |- _ ] => destruct E |

adam@351 | 183 | [ _ : S (length ?E) < 2 |- _ ] => destruct E |

adam@480 | 184 | [ IH : _ |- context[split ?L] ] => |

adam@480 | 185 specialize (IH L); destruct (split L); destruct IH |

adam@351 | 186 end; crush). |

adam@351 | 187 Defined. |

adam@351 | 188 |

adam@480 | 189 Ltac split_wf := intros ls ?; intros; generalize (@split_wf (length ls) ls); |

adam@480 | 190 destruct (split ls); destruct 1; crush. |

adam@351 | 191 |

adam@480 | 192 Lemma split_wf1 : forall ls, 2 <= length ls |

adam@480 | 193 -> lengthOrder (fst (split ls)) ls. |

adam@480 | 194 split_wf. |

adam@351 | 195 Defined. |

adam@351 | 196 |

adam@480 | 197 Lemma split_wf2 : forall ls, 2 <= length ls |

adam@480 | 198 -> lengthOrder (snd (split ls)) ls. |

adam@480 | 199 split_wf. |

adam@351 | 200 Defined. |

adam@351 | 201 |

adam@480 | 202 Hint Resolve split_wf1 split_wf2. |

adam@351 | 203 |

adam@453 | 204 (** 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 | 205 |

adam@351 | 206 Definition mergeSort : list A -> list A. |

adam@351 | 207 (* begin thide *) |

adam@351 | 208 refine (Fix lengthOrder_wf (fun _ => list A) |

adam@351 | 209 (fun (ls : list A) |

adam@351 | 210 (mergeSort : forall ls' : list A, lengthOrder ls' ls -> list A) => |

adam@351 | 211 if le_lt_dec 2 (length ls) |

adam@480 | 212 then let lss := split ls in |

adam@351 | 213 merge (mergeSort (fst lss) _) (mergeSort (snd lss) _) |

adam@351 | 214 else ls)); subst lss; eauto. |

adam@351 | 215 Defined. |

adam@351 | 216 (* end thide *) |

adam@351 | 217 End mergeSort. |

adam@351 | 218 |

adam@351 | 219 (** The important thing is that it is now easy to evaluate calls to [mergeSort]. *) |

adam@351 | 220 |

adam@351 | 221 Eval compute in mergeSort leb (1 :: 2 :: 36 :: 8 :: 19 :: nil). |

adam@351 | 222 (** [= 1 :: 2 :: 8 :: 19 :: 36 :: nil] *) |

adam@351 | 223 |

adam@441 | 224 (** %\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 | 225 |

adam@351 | 226 (* begin thide *) |

adam@351 | 227 Theorem mergeSort_eq : forall A (le : A -> A -> bool) ls, |

adam@351 | 228 mergeSort le ls = if le_lt_dec 2 (length ls) |

adam@480 | 229 then let lss := split ls in |

adam@351 | 230 merge le (mergeSort le (fst lss)) (mergeSort le (snd lss)) |

adam@351 | 231 else ls. |

adam@351 | 232 intros; apply (Fix_eq (@lengthOrder_wf A) (fun _ => list A)); intros. |

adam@351 | 233 |

adam@424 | 234 (** 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 | 235 |

adam@351 | 236 Check Fix_eq. |

adam@351 | 237 (** %\vspace{-.15in}%[[ |

adam@351 | 238 Fix_eq |

adam@351 | 239 : forall (A : Type) (R : A -> A -> Prop) (Rwf : well_founded R) |

adam@351 | 240 (P : A -> Type) |

adam@351 | 241 (F : forall x : A, (forall y : A, R y x -> P y) -> P x), |

adam@351 | 242 (forall (x : A) (f g : forall y : A, R y x -> P y), |

adam@351 | 243 (forall (y : A) (p : R y x), f y p = g y p) -> F x f = F x g) -> |

adam@351 | 244 forall x : A, |

adam@351 | 245 Fix Rwf P F x = F x (fun (y : A) (_ : R y x) => Fix Rwf P F y) |

adam@351 | 246 ]] |

adam@351 | 247 |

adam@465 | 248 Most such obligations are dischargeable with straightforward proof automation, and this example is no exception. *) |

adam@351 | 249 |

adam@351 | 250 match goal with |

adam@351 | 251 | [ |- context[match ?E with left _ => _ | right _ => _ end] ] => destruct E |

adam@351 | 252 end; simpl; f_equal; auto. |

adam@351 | 253 Qed. |

adam@351 | 254 (* end thide *) |

adam@351 | 255 |

adam@351 | 256 (** As a final test of our definition's suitability, we can extract to OCaml. *) |

adam@351 | 257 |

adam@351 | 258 Extraction mergeSort. |

adam@351 | 259 |

adam@351 | 260 (** << |

adam@351 | 261 let rec mergeSort le x = |

adam@351 | 262 match le_lt_dec (S (S O)) (length x) with |

adam@351 | 263 | Left -> |

adam@480 | 264 let lss = split x in |

adam@351 | 265 merge le (mergeSort le (fst lss)) (mergeSort le (snd lss)) |

adam@351 | 266 | Right -> x |

adam@351 | 267 >> |

adam@351 | 268 |

adam@353 | 269 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 | 270 |

adam@351 | 271 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 | 272 |

adam@351 | 273 Check well_founded_induction. |

adam@351 | 274 (** %\vspace{-.15in}%[[ |

adam@351 | 275 well_founded_induction |

adam@351 | 276 : forall (A : Type) (R : A -> A -> Prop), |

adam@351 | 277 well_founded R -> |

adam@351 | 278 forall P : A -> Set, |

adam@351 | 279 (forall x : A, (forall y : A, R y x -> P y) -> P x) -> |

adam@351 | 280 forall a : A, P a |

adam@351 | 281 ]] |

adam@351 | 282 |

adam@351 | 283 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 | 284 |

adam@352 | 285 |

adam@354 | 286 (** * A Non-Termination Monad Inspired by Domain Theory *) |

adam@352 | 287 |

adam@424 | 288 (** 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 | 289 |

adam@355 | 290 Consider this definition of a type of computations. *) |

adam@355 | 291 |

adam@352 | 292 Section computation. |

adam@352 | 293 Variable A : Type. |

adam@355 | 294 (** The type [A] describes the result a computation will yield, if it terminates. |

adam@355 | 295 |

adam@355 | 296 We give a rich dependent type to computations themselves: *) |

adam@352 | 297 |

adam@352 | 298 Definition computation := |

adam@352 | 299 {f : nat -> option A |

adam@352 | 300 | forall (n : nat) (v : A), |

adam@352 | 301 f n = Some v |

adam@352 | 302 -> forall (n' : nat), n' >= n |

adam@352 | 303 -> f n' = Some v}. |

adam@352 | 304 |

adam@474 | 305 (** 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 | 306 |

adam@355 | 307 It is easy to define a relation characterizing when a computation runs to a particular result at a particular approximation level. *) |

adam@355 | 308 |

adam@352 | 309 Definition runTo (m : computation) (n : nat) (v : A) := |

adam@352 | 310 proj1_sig m n = Some v. |

adam@352 | 311 |

adam@355 | 312 (** On top of [runTo], we also define [run], which is the most abstract notion of when a computation runs to a value. *) |

adam@355 | 313 |

adam@352 | 314 Definition run (m : computation) (v : A) := |

adam@352 | 315 exists n, runTo m n v. |

adam@352 | 316 End computation. |

adam@352 | 317 |

adam@355 | 318 (** 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 | 319 (* begin hide *) |

adam@355 | 320 |

adam@352 | 321 Hint Unfold runTo. |

adam@352 | 322 |

adam@352 | 323 Ltac run' := unfold run, runTo in *; try red; crush; |

adam@352 | 324 repeat (match goal with |

adam@352 | 325 | [ _ : proj1_sig ?E _ = _ |- _ ] => |

adam@352 | 326 match goal with |

adam@352 | 327 | [ x : _ |- _ ] => |

adam@352 | 328 match x with |

adam@352 | 329 | E => destruct E |

adam@352 | 330 end |

adam@352 | 331 end |

adam@352 | 332 | [ |- context[match ?M with exist _ _ => _ end] ] => let Heq := fresh "Heq" in |

adam@352 | 333 case_eq M; intros ? ? Heq; try rewrite Heq in *; try subst |

adam@352 | 334 | [ _ : context[match ?M with exist _ _ => _ end] |- _ ] => let Heq := fresh "Heq" in |

adam@352 | 335 case_eq M; intros ? ? Heq; try rewrite Heq in *; subst |

adam@352 | 336 | [ H : forall n v, ?E n = Some v -> _, |

adam@352 | 337 _ : context[match ?E ?N with Some _ => _ | None => _ end] |- _ ] => |

adam@426 | 338 specialize (H N); destruct (E N); try rewrite (H _ (eq_refl _)) by auto; try discriminate |

adam@352 | 339 | [ H : forall n v, ?E n = Some v -> _, H' : ?E _ = Some _ |- _ ] => rewrite (H _ _ H') by auto |

adam@352 | 340 end; simpl in *); eauto 7. |

adam@352 | 341 |

adam@352 | 342 Ltac run := run'; repeat (match goal with |

adam@352 | 343 | [ H : forall n v, ?E n = Some v -> _ |

adam@352 | 344 |- context[match ?E ?N with Some _ => _ | None => _ end] ] => |

adam@426 | 345 specialize (H N); destruct (E N); try rewrite (H _ (eq_refl _)) by auto; try discriminate |

adam@352 | 346 end; run'). |

adam@352 | 347 |

adam@352 | 348 Lemma ex_irrelevant : forall P : Prop, P -> exists n : nat, P. |

adam@352 | 349 exists 0; auto. |

adam@352 | 350 Qed. |

adam@352 | 351 |

adam@352 | 352 Hint Resolve ex_irrelevant. |

adam@352 | 353 |

adam@352 | 354 Require Import Max. |

adam@352 | 355 |

adam@380 | 356 Theorem max_spec_le : forall n m, n <= m /\ max n m = m \/ m <= n /\ max n m = n. |

adam@380 | 357 induction n; destruct m; simpl; intuition; |

adam@380 | 358 specialize (IHn m); intuition. |

adam@380 | 359 Qed. |

adam@380 | 360 |

adam@352 | 361 Ltac max := intros n m; generalize (max_spec_le n m); crush. |

adam@352 | 362 |

adam@352 | 363 Lemma max_1 : forall n m, max n m >= n. |

adam@352 | 364 max. |

adam@352 | 365 Qed. |

adam@352 | 366 |

adam@352 | 367 Lemma max_2 : forall n m, max n m >= m. |

adam@352 | 368 max. |

adam@352 | 369 Qed. |

adam@352 | 370 |

adam@352 | 371 Hint Resolve max_1 max_2. |

adam@352 | 372 |

adam@352 | 373 Lemma ge_refl : forall n, n >= n. |

adam@352 | 374 crush. |

adam@352 | 375 Qed. |

adam@352 | 376 |

adam@352 | 377 Hint Resolve ge_refl. |

adam@352 | 378 |

adam@352 | 379 Hint Extern 1 => match goal with |

adam@352 | 380 | [ H : _ = exist _ _ _ |- _ ] => rewrite H |

adam@352 | 381 end. |

adam@355 | 382 (* end hide *) |

adam@355 | 383 (** remove printing exists *) |

adam@355 | 384 |

adam@480 | 385 (** 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 | 386 |

adam@352 | 387 Section Bottom. |

adam@352 | 388 Variable A : Type. |

adam@352 | 389 |

adam@352 | 390 Definition Bottom : computation A. |

adam@352 | 391 exists (fun _ : nat => @None A); abstract run. |

adam@352 | 392 Defined. |

adam@352 | 393 |

adam@352 | 394 Theorem run_Bottom : forall v, ~run Bottom v. |

adam@352 | 395 run. |

adam@352 | 396 Qed. |

adam@352 | 397 End Bottom. |

adam@352 | 398 |

adam@355 | 399 (** A slightly more complicated example is [Return], which gives the same terminating answer at every approximation level. *) |

adam@355 | 400 |

adam@352 | 401 Section Return. |

adam@352 | 402 Variable A : Type. |

adam@352 | 403 Variable v : A. |

adam@352 | 404 |

adam@352 | 405 Definition Return : computation A. |

adam@352 | 406 intros; exists (fun _ : nat => Some v); abstract run. |

adam@352 | 407 Defined. |

adam@352 | 408 |

adam@352 | 409 Theorem run_Return : run Return v. |

adam@352 | 410 run. |

adam@352 | 411 Qed. |

adam@352 | 412 End Return. |

adam@352 | 413 |

adam@474 | 414 (** 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 | 415 |

adam@352 | 416 Section Bind. |

adam@352 | 417 Variables A B : Type. |

adam@352 | 418 Variable m1 : computation A. |

adam@352 | 419 Variable m2 : A -> computation B. |

adam@352 | 420 |

adam@352 | 421 Definition Bind : computation B. |

adam@352 | 422 exists (fun n => |

adam@357 | 423 let (f1, _) := m1 in |

adam@352 | 424 match f1 n with |

adam@352 | 425 | None => None |

adam@352 | 426 | Some v => |

adam@357 | 427 let (f2, _) := m2 v in |

adam@352 | 428 f2 n |

adam@352 | 429 end); abstract run. |

adam@352 | 430 Defined. |

adam@352 | 431 |

adam@352 | 432 Theorem run_Bind : forall (v1 : A) (v2 : B), |

adam@352 | 433 run m1 v1 |

adam@352 | 434 -> run (m2 v1) v2 |

adam@352 | 435 -> run Bind v2. |

adam@352 | 436 run; match goal with |

adam@352 | 437 | [ x : nat, y : nat |- _ ] => exists (max x y) |

adam@352 | 438 end; run. |

adam@352 | 439 Qed. |

adam@352 | 440 End Bind. |

adam@352 | 441 |

adam@355 | 442 (** A simple notation lets us write [Bind] calls the way they appear in Haskell. *) |

adam@352 | 443 |

adam@352 | 444 Notation "x <- m1 ; m2" := |

adam@352 | 445 (Bind m1 (fun x => m2)) (right associativity, at level 70). |

adam@352 | 446 |

adam@424 | 447 (** 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 | 448 |

adam@352 | 449 Definition meq A (m1 m2 : computation A) := forall n, proj1_sig m1 n = proj1_sig m2 n. |

adam@352 | 450 |

adam@352 | 451 Theorem left_identity : forall A B (a : A) (f : A -> computation B), |

adam@352 | 452 meq (Bind (Return a) f) (f a). |

adam@352 | 453 run. |

adam@352 | 454 Qed. |

adam@352 | 455 |

adam@352 | 456 Theorem right_identity : forall A (m : computation A), |

adam@352 | 457 meq (Bind m (@Return _)) m. |

adam@352 | 458 run. |

adam@352 | 459 Qed. |

adam@352 | 460 |

adam@357 | 461 Theorem associativity : forall A B C (m : computation A) |

adam@357 | 462 (f : A -> computation B) (g : B -> computation C), |

adam@352 | 463 meq (Bind (Bind m f) g) (Bind m (fun x => Bind (f x) g)). |

adam@352 | 464 run. |

adam@352 | 465 Qed. |

adam@352 | 466 |

adam@398 | 467 (** 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 | 468 |

adam@352 | 469 Section lattice. |

adam@352 | 470 Variable A : Type. |

adam@352 | 471 |

adam@352 | 472 Definition leq (x y : option A) := |

adam@352 | 473 forall v, x = Some v -> y = Some v. |

adam@352 | 474 End lattice. |

adam@352 | 475 |

adam@355 | 476 (** 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 | 477 |

adam@352 | 478 Section Fix. |

adam@475 | 479 |

adam@355 | 480 (** First, we have the function domain and range types. *) |

adam@355 | 481 |

adam@352 | 482 Variables A B : Type. |

adam@355 | 483 |

adam@355 | 484 (** Next comes the function body, which is written as though it can be parameterized over itself, for recursive calls. *) |

adam@355 | 485 |

adam@352 | 486 Variable f : (A -> computation B) -> (A -> computation B). |

adam@352 | 487 |

adam@355 | 488 (** 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 | 489 |

adam@352 | 490 Hypothesis f_continuous : forall n v v1 x, |

adam@352 | 491 runTo (f v1 x) n v |

adam@352 | 492 -> forall (v2 : A -> computation B), |

adam@352 | 493 (forall x, leq (proj1_sig (v1 x) n) (proj1_sig (v2 x) n)) |

adam@352 | 494 -> runTo (f v2 x) n v. |

adam@352 | 495 |

adam@355 | 496 (** 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 | 497 |

adam@352 | 498 Fixpoint Fix' (n : nat) (x : A) : computation B := |

adam@352 | 499 match n with |

adam@352 | 500 | O => Bottom _ |

adam@352 | 501 | S n' => f (Fix' n') x |

adam@352 | 502 end. |

adam@352 | 503 |

adam@355 | 504 (** Now it is straightforward to package [Fix'] as a computation combinator [Fix]. *) |

adam@355 | 505 |

adam@352 | 506 Hint Extern 1 (_ >= _) => omega. |

adam@352 | 507 Hint Unfold leq. |

adam@352 | 508 |

adam@352 | 509 Lemma Fix'_ok : forall steps n x v, proj1_sig (Fix' n x) steps = Some v |

adam@352 | 510 -> forall n', n' >= n |

adam@352 | 511 -> proj1_sig (Fix' n' x) steps = Some v. |

adam@352 | 512 unfold runTo in *; induction n; crush; |

adam@352 | 513 match goal with |

adam@352 | 514 | [ H : _ >= _ |- _ ] => inversion H; crush; eauto |

adam@352 | 515 end. |

adam@352 | 516 Qed. |

adam@352 | 517 |

adam@352 | 518 Hint Resolve Fix'_ok. |

adam@352 | 519 |

adam@352 | 520 Hint Extern 1 (proj1_sig _ _ = _) => simpl; |

adam@352 | 521 match goal with |

adam@352 | 522 | [ |- proj1_sig ?E _ = _ ] => eapply (proj2_sig E) |

adam@352 | 523 end. |

adam@352 | 524 |

adam@352 | 525 Definition Fix : A -> computation B. |

adam@352 | 526 intro x; exists (fun n => proj1_sig (Fix' n x) n); abstract run. |

adam@352 | 527 Defined. |

adam@352 | 528 |

adam@355 | 529 (** Finally, we can prove that [Fix] obeys the expected computation rule. *) |

adam@352 | 530 |

adam@352 | 531 Theorem run_Fix : forall x v, |

adam@352 | 532 run (f Fix x) v |

adam@352 | 533 -> run (Fix x) v. |

adam@352 | 534 run; match goal with |

adam@352 | 535 | [ n : nat |- _ ] => exists (S n); eauto |

adam@352 | 536 end. |

adam@352 | 537 Qed. |

adam@352 | 538 End Fix. |

adam@352 | 539 |

adam@355 | 540 (* begin hide *) |

adam@352 | 541 Lemma leq_Some : forall A (x y : A), leq (Some x) (Some y) |

adam@352 | 542 -> x = y. |

adam@426 | 543 intros ? ? ? H; generalize (H _ (eq_refl _)); crush. |

adam@352 | 544 Qed. |

adam@352 | 545 |

adam@352 | 546 Lemma leq_None : forall A (x y : A), leq (Some x) None |

adam@352 | 547 -> False. |

adam@426 | 548 intros ? ? ? H; generalize (H _ (eq_refl _)); crush. |

adam@352 | 549 Qed. |

adam@352 | 550 |

adam@355 | 551 Ltac mergeSort' := run; |

adam@355 | 552 repeat (match goal with |

adam@355 | 553 | [ |- context[match ?E with O => _ | S _ => _ end] ] => destruct E |

adam@355 | 554 end; run); |

adam@355 | 555 repeat match goal with |

adam@355 | 556 | [ H : forall x, leq (proj1_sig (?f x) _) (proj1_sig (?g x) _) |- _ ] => |

adam@355 | 557 match goal with |

adam@355 | 558 | [ H1 : f ?arg = _, H2 : g ?arg = _ |- _ ] => |

adam@355 | 559 generalize (H arg); rewrite H1; rewrite H2; clear H1 H2; simpl; intro |

adam@355 | 560 end |

adam@355 | 561 end; run; repeat match goal with |

adam@355 | 562 | [ H : _ |- _ ] => (apply leq_None in H; tauto) || (apply leq_Some in H; subst) |

adam@355 | 563 end; auto. |

adam@355 | 564 (* end hide *) |

adam@355 | 565 |

adam@355 | 566 (** 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 | 567 |

adam@352 | 568 Definition mergeSort' : forall A, (A -> A -> bool) -> list A -> computation (list A). |

adam@352 | 569 refine (fun A le => Fix |

adam@352 | 570 (fun (mergeSort : list A -> computation (list A)) |

adam@352 | 571 (ls : list A) => |

adam@352 | 572 if le_lt_dec 2 (length ls) |

adam@480 | 573 then let lss := split ls in |

adam@352 | 574 ls1 <- mergeSort (fst lss); |

adam@352 | 575 ls2 <- mergeSort (snd lss); |

adam@352 | 576 Return (merge le ls1 ls2) |

adam@355 | 577 else Return ls) _); abstract mergeSort'. |

adam@352 | 578 Defined. |

adam@352 | 579 |

adam@424 | 580 (** 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 | 581 |

adam@352 | 582 Lemma test_mergeSort' : run (mergeSort' leb (1 :: 2 :: 36 :: 8 :: 19 :: nil)) |

adam@352 | 583 (1 :: 2 :: 8 :: 19 :: 36 :: nil). |

adam@352 | 584 exists 4; reflexivity. |

adam@352 | 585 Qed. |

adam@352 | 586 |

adam@453 | 587 (** 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 | 588 |

adam@355 | 589 (* begin hide *) |

adam@355 | 590 Ltac looper := unfold leq in *; run; |

adam@355 | 591 repeat match goal with |

adam@355 | 592 | [ x : unit |- _ ] => destruct x |

adam@355 | 593 | [ x : bool |- _ ] => destruct x |

adam@355 | 594 end; auto. |

adam@355 | 595 (* end hide *) |

adam@355 | 596 |

adam@352 | 597 Definition looper : bool -> computation unit. |

adam@352 | 598 refine (Fix (fun looper (b : bool) => |

adam@355 | 599 if b then Return tt else looper b) _); abstract looper. |

adam@352 | 600 Defined. |

adam@352 | 601 |

adam@352 | 602 Lemma test_looper : run (looper true) tt. |

adam@352 | 603 exists 1; reflexivity. |

adam@352 | 604 Qed. |

adam@354 | 605 |

adam@355 | 606 (** As before, proving outputs for specific inputs is as easy as demonstrating a high enough approximation level. |

adam@355 | 607 |

adam@480 | 608 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 | 609 |

adam@354 | 610 |

adam@354 | 611 (** * Co-Inductive Non-Termination Monads *) |

adam@354 | 612 |

adam@356 | 613 (** 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 | 614 |

adam@356 | 615 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 | 616 |

adam@354 | 617 CoInductive thunk (A : Type) : Type := |

adam@354 | 618 | Answer : A -> thunk A |

adam@354 | 619 | Think : thunk A -> thunk A. |

adam@354 | 620 |

adam@356 | 621 (** 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 | 622 |

adam@424 | 623 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 | 624 |

adam@356 | 625 CoFixpoint TBind A B (m1 : thunk A) (m2 : A -> thunk B) : thunk B := |

adam@354 | 626 match m1 with |

adam@354 | 627 | Answer x => m2 x |

adam@354 | 628 | Think m1' => Think (TBind m1' m2) |

adam@354 | 629 end. |

adam@354 | 630 |

adam@356 | 631 (** 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 | 632 |

adam@356 | 633 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 | 634 |

adam@356 | 635 (* begin hide *) |

adam@354 | 636 CoInductive thunk_eq A : thunk A -> thunk A -> Prop := |

adam@354 | 637 | EqAnswer : forall x, thunk_eq (Answer x) (Answer x) |

adam@354 | 638 | EqThinkL : forall m1 m2, thunk_eq m1 m2 -> thunk_eq (Think m1) m2 |

adam@354 | 639 | EqThinkR : forall m1 m2, thunk_eq m1 m2 -> thunk_eq m1 (Think m2). |

adam@354 | 640 |

adam@354 | 641 Section thunk_eq_coind. |

adam@354 | 642 Variable A : Type. |

adam@354 | 643 Variable P : thunk A -> thunk A -> Prop. |

adam@354 | 644 |

adam@354 | 645 Hypothesis H : forall m1 m2, P m1 m2 |

adam@354 | 646 -> match m1, m2 with |

adam@354 | 647 | Answer x1, Answer x2 => x1 = x2 |

adam@354 | 648 | Think m1', Think m2' => P m1' m2' |

adam@354 | 649 | Think m1', _ => P m1' m2 |

adam@354 | 650 | _, Think m2' => P m1 m2' |

adam@354 | 651 end. |

adam@354 | 652 |

adam@354 | 653 Theorem thunk_eq_coind : forall m1 m2, P m1 m2 -> thunk_eq m1 m2. |

adam@354 | 654 cofix; intros; |

adam@354 | 655 match goal with |

adam@354 | 656 | [ H' : P _ _ |- _ ] => specialize (H H'); clear H' |

adam@354 | 657 end; destruct m1; destruct m2; subst; repeat constructor; auto. |

adam@354 | 658 Qed. |

adam@354 | 659 End thunk_eq_coind. |

adam@356 | 660 (* end hide *) |

adam@356 | 661 |

adam@356 | 662 (** 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 | 663 |

adam@354 | 664 Definition frob A (m : thunk A) : thunk A := |

adam@354 | 665 match m with |

adam@354 | 666 | Answer x => Answer x |

adam@354 | 667 | Think m' => Think m' |

adam@354 | 668 end. |

adam@354 | 669 |

adam@354 | 670 Theorem frob_eq : forall A (m : thunk A), frob m = m. |

adam@354 | 671 destruct m; reflexivity. |

adam@354 | 672 Qed. |

adam@354 | 673 |

adam@356 | 674 (* begin hide *) |

adam@354 | 675 Theorem thunk_eq_frob : forall A (m1 m2 : thunk A), |

adam@354 | 676 thunk_eq (frob m1) (frob m2) |

adam@354 | 677 -> thunk_eq m1 m2. |

adam@354 | 678 intros; repeat rewrite frob_eq in *; auto. |

adam@354 | 679 Qed. |

adam@354 | 680 |

adam@354 | 681 Ltac findDestr := match goal with |

adam@354 | 682 | [ |- context[match ?E with Answer _ => _ | Think _ => _ end] ] => |

adam@354 | 683 match E with |

adam@354 | 684 | context[match _ with Answer _ => _ | Think _ => _ end] => fail 1 |

adam@354 | 685 | _ => destruct E |

adam@354 | 686 end |

adam@354 | 687 end. |

adam@354 | 688 |

adam@354 | 689 Theorem thunk_eq_refl : forall A (m : thunk A), thunk_eq m m. |

adam@354 | 690 intros; apply (thunk_eq_coind (fun m1 m2 => m1 = m2)); crush; findDestr; reflexivity. |

adam@354 | 691 Qed. |

adam@354 | 692 |

adam@354 | 693 Hint Resolve thunk_eq_refl. |

adam@354 | 694 |

adam@354 | 695 Theorem tleft_identity : forall A B (a : A) (f : A -> thunk B), |

adam@354 | 696 thunk_eq (TBind (Answer a) f) (f a). |

adam@354 | 697 intros; apply thunk_eq_frob; crush. |

adam@354 | 698 Qed. |

adam@354 | 699 |

adam@354 | 700 Theorem tright_identity : forall A (m : thunk A), |

adam@354 | 701 thunk_eq (TBind m (@Answer _)) m. |

adam@354 | 702 intros; apply (thunk_eq_coind (fun m1 m2 => m1 = TBind m2 (@Answer _))); crush; |

adam@354 | 703 findDestr; reflexivity. |

adam@354 | 704 Qed. |

adam@354 | 705 |

adam@354 | 706 Lemma TBind_Answer : forall (A B : Type) (v : A) (m2 : A -> thunk B), |

adam@354 | 707 TBind (Answer v) m2 = m2 v. |

adam@354 | 708 intros; rewrite <- (frob_eq (TBind (Answer v) m2)); |

adam@354 | 709 simpl; findDestr; reflexivity. |

adam@354 | 710 Qed. |

adam@354 | 711 |

adam@375 | 712 Hint Rewrite TBind_Answer. |

adam@354 | 713 |

adam@355 | 714 (** printing exists $\exists$ *) |

adam@355 | 715 |

adam@354 | 716 Theorem tassociativity : forall A B C (m : thunk A) (f : A -> thunk B) (g : B -> thunk C), |

adam@354 | 717 thunk_eq (TBind (TBind m f) g) (TBind m (fun x => TBind (f x) g)). |

adam@354 | 718 intros; apply (thunk_eq_coind (fun m1 m2 => (exists m, |

adam@354 | 719 m1 = TBind (TBind m f) g |

adam@354 | 720 /\ m2 = TBind m (fun x => TBind (f x) g)) |

adam@354 | 721 \/ m1 = m2)); crush; eauto; repeat (findDestr; crush; eauto). |

adam@354 | 722 Qed. |

adam@356 | 723 (* end hide *) |

adam@356 | 724 |

adam@356 | 725 (** As a simple example, here is how we might define a tail-recursive factorial function. *) |

adam@354 | 726 |

adam@354 | 727 CoFixpoint fact (n acc : nat) : thunk nat := |

adam@354 | 728 match n with |

adam@354 | 729 | O => Answer acc |

adam@354 | 730 | S n' => Think (fact n' (S n' * acc)) |

adam@354 | 731 end. |

adam@354 | 732 |

adam@356 | 733 (** To test our definition, we need an evaluation relation that characterizes results of evaluating [thunk]s. *) |

adam@356 | 734 |

adam@354 | 735 Inductive eval A : thunk A -> A -> Prop := |

adam@354 | 736 | EvalAnswer : forall x, eval (Answer x) x |

adam@354 | 737 | EvalThink : forall m x, eval m x -> eval (Think m) x. |

adam@354 | 738 |

adam@375 | 739 Hint Rewrite frob_eq. |

adam@354 | 740 |

adam@354 | 741 Lemma eval_frob : forall A (c : thunk A) x, |

adam@354 | 742 eval (frob c) x |

adam@354 | 743 -> eval c x. |

adam@354 | 744 crush. |

adam@354 | 745 Qed. |

adam@354 | 746 |

adam@354 | 747 Theorem eval_fact : eval (fact 5 1) 120. |

adam@354 | 748 repeat (apply eval_frob; simpl; constructor). |

adam@354 | 749 Qed. |

adam@354 | 750 |

adam@356 | 751 (** We need to apply constructors of [eval] explicitly, but the process is easy to automate completely for concrete input programs. |

adam@356 | 752 |

adam@465 | 753 Now consider another very similar definition, this time of a Fibonacci number function. *) |

adam@357 | 754 |

adam@357 | 755 Notation "x <- m1 ; m2" := |

adam@357 | 756 (TBind m1 (fun x => m2)) (right associativity, at level 70). |

adam@357 | 757 |

adam@404 | 758 (* begin hide *) |

adam@437 | 759 (* begin thide *) |

adam@424 | 760 Definition fib := pred. |

adam@437 | 761 (* end thide *) |

adam@404 | 762 (* end hide *) |

adam@404 | 763 |

adam@453 | 764 (** %\vspace{-.3in}%[[ |

adam@354 | 765 CoFixpoint fib (n : nat) : thunk nat := |

adam@354 | 766 match n with |

adam@354 | 767 | 0 => Answer 1 |

adam@354 | 768 | 1 => Answer 1 |

adam@357 | 769 | _ => n1 <- fib (pred n); |

adam@357 | 770 n2 <- fib (pred (pred n)); |

adam@357 | 771 Answer (n1 + n2) |

adam@354 | 772 end. |

adam@354 | 773 ]] |

adam@354 | 774 |

adam@356 | 775 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 | 776 |

adam@356 | 777 %\medskip% |

adam@356 | 778 |

adam@424 | 779 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 | 780 |

adam@354 | 781 CoInductive comp (A : Type) : Type := |

adam@354 | 782 | Ret : A -> comp A |

adam@354 | 783 | Bnd : forall B, comp B -> (B -> comp A) -> comp A. |

adam@354 | 784 |

adam@404 | 785 (** 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 | 786 |

adam@356 | 787 It is easy to define the semantics of terminating [comp] computations. *) |

adam@356 | 788 |

adam@354 | 789 Inductive exec A : comp A -> A -> Prop := |

adam@354 | 790 | ExecRet : forall x, exec (Ret x) x |

adam@354 | 791 | ExecBnd : forall B (c : comp B) (f : B -> comp A) x1 x2, exec (A := B) c x1 |

adam@354 | 792 -> exec (f x1) x2 |

adam@354 | 793 -> exec (Bnd c f) x2. |

adam@354 | 794 |

adam@356 | 795 (** 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 | 796 |

adam@356 | 797 (* begin hide *) |

adam@354 | 798 Hint Constructors exec. |

adam@354 | 799 |

adam@354 | 800 Definition comp_eq A (c1 c2 : comp A) := forall r, exec c1 r <-> exec c2 r. |

adam@354 | 801 |

adam@354 | 802 Ltac inverter := repeat match goal with |

adam@354 | 803 | [ H : exec _ _ |- _ ] => inversion H; []; crush |

adam@354 | 804 end. |

adam@354 | 805 |

adam@354 | 806 Theorem cleft_identity : forall A B (a : A) (f : A -> comp B), |

adam@354 | 807 comp_eq (Bnd (Ret a) f) (f a). |

adam@354 | 808 red; crush; inverter; eauto. |

adam@354 | 809 Qed. |

adam@354 | 810 |

adam@354 | 811 Theorem cright_identity : forall A (m : comp A), |

adam@354 | 812 comp_eq (Bnd m (@Ret _)) m. |

adam@354 | 813 red; crush; inverter; eauto. |

adam@354 | 814 Qed. |

adam@354 | 815 |

adam@354 | 816 Lemma cassociativity1 : forall A B C (f : A -> comp B) (g : B -> comp C) r c, |

adam@354 | 817 exec c r |

adam@354 | 818 -> forall m, c = Bnd (Bnd m f) g |

adam@354 | 819 -> exec (Bnd m (fun x => Bnd (f x) g)) r. |

adam@354 | 820 induction 1; crush. |

adam@354 | 821 match goal with |

adam@354 | 822 | [ H : Bnd _ _ = Bnd _ _ |- _ ] => injection H; clear H; intros; try subst |

adam@354 | 823 end. |

adam@354 | 824 move H3 after A. |

adam@354 | 825 generalize dependent B0. |

adam@354 | 826 do 2 intro. |

adam@354 | 827 subst. |

adam@354 | 828 crush. |

adam@354 | 829 inversion H; clear H; crush. |

adam@354 | 830 eauto. |

adam@354 | 831 Qed. |

adam@354 | 832 |

adam@354 | 833 Lemma cassociativity2 : forall A B C (f : A -> comp B) (g : B -> comp C) r c, |

adam@354 | 834 exec c r |

adam@354 | 835 -> forall m, c = Bnd m (fun x => Bnd (f x) g) |

adam@354 | 836 -> exec (Bnd (Bnd m f) g) r. |

adam@354 | 837 induction 1; crush. |

adam@354 | 838 match goal with |

adam@354 | 839 | [ H : Bnd _ _ = Bnd _ _ |- _ ] => injection H; clear H; intros; try subst |

adam@354 | 840 end. |

adam@354 | 841 move H3 after B. |

adam@354 | 842 generalize dependent B0. |

adam@354 | 843 do 2 intro. |

adam@354 | 844 subst. |

adam@354 | 845 crush. |

adam@354 | 846 inversion H0; clear H0; crush. |

adam@354 | 847 eauto. |

adam@354 | 848 Qed. |

adam@354 | 849 |

adam@354 | 850 Hint Resolve cassociativity1 cassociativity2. |

adam@354 | 851 |

adam@354 | 852 Theorem cassociativity : forall A B C (m : comp A) (f : A -> comp B) (g : B -> comp C), |

adam@354 | 853 comp_eq (Bnd (Bnd m f) g) (Bnd m (fun x => Bnd (f x) g)). |

adam@354 | 854 red; crush; eauto. |

adam@354 | 855 Qed. |

adam@356 | 856 (* end hide *) |

adam@356 | 857 |

adam@469 | 858 (** 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 | 859 |

adam@356 | 860 Notation "x <- m1 ; m2" := (Bnd m1 (fun x => m2)). |

adam@354 | 861 |

adam@354 | 862 CoFixpoint mergeSort'' A (le : A -> A -> bool) (ls : list A) : comp (list A) := |

adam@354 | 863 if le_lt_dec 2 (length ls) |

adam@480 | 864 then let lss := split ls in |

adam@356 | 865 ls1 <- mergeSort'' le (fst lss); |

adam@356 | 866 ls2 <- mergeSort'' le (snd lss); |

adam@356 | 867 Ret (merge le ls1 ls2) |

adam@354 | 868 else Ret ls. |

adam@354 | 869 |

adam@356 | 870 (** To execute this function, we go through the usual exercise of writing a function to catalyze evaluation of co-recursive calls. *) |

adam@356 | 871 |

adam@354 | 872 Definition frob' A (c : comp A) := |

adam@354 | 873 match c with |

adam@354 | 874 | Ret x => Ret x |

adam@354 | 875 | Bnd _ c' f => Bnd c' f |

adam@354 | 876 end. |

adam@354 | 877 |

adam@354 | 878 Lemma exec_frob : forall A (c : comp A) x, |

adam@354 | 879 exec (frob' c) x |

adam@354 | 880 -> exec c x. |

adam@356 | 881 destruct c; crush. |

adam@354 | 882 Qed. |

adam@354 | 883 |

adam@356 | 884 (** Now the same sort of proof script that we applied for testing [thunk]s will get the job done. *) |

adam@356 | 885 |

adam@354 | 886 Lemma test_mergeSort'' : exec (mergeSort'' leb (1 :: 2 :: 36 :: 8 :: 19 :: nil)) |

adam@354 | 887 (1 :: 2 :: 8 :: 19 :: 36 :: nil). |

adam@354 | 888 repeat (apply exec_frob; simpl; econstructor). |

adam@354 | 889 Qed. |

adam@354 | 890 |

adam@356 | 891 (** 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 | 892 |

adam@354 | 893 Definition curriedAdd (n : nat) := Ret (fun m : nat => Ret (n + m)). |

adam@354 | 894 |

adam@356 | 895 (** This definition works fine, but we run into trouble when we try to apply it in a trivial way. |

adam@356 | 896 [[ |

adam@356 | 897 Definition testCurriedAdd := Bnd (curriedAdd 2) (fun f => f 3). |

adam@354 | 898 ]] |

adam@354 | 899 |

adam@354 | 900 << |

adam@354 | 901 Error: Universe inconsistency. |

adam@354 | 902 >> |

adam@356 | 903 |

adam@496 | 904 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 | 905 |

adam@354 | 906 |

adam@357 | 907 (** * Comparing the Alternatives *) |

adam@354 | 908 |

adam@453 | 909 (** 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 | 910 |

adam@356 | 911 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 | 912 |

adam@356 | 913 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 | 914 |

adam@480 | 915 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 | 916 |

adam@465 | 917 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 | 918 |

adam@356 | 919 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 | 920 |

adam@356 | 921 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 | 922 |

adam@354 | 923 Definition curriedAdd' (n : nat) := Return (fun m : nat => Return (n + m)). |

adam@354 | 924 |

adam@356 | 925 Definition testCurriedAdd := Bind (curriedAdd' 2) (fun f => f 3). |

adam@356 | 926 |

adam@357 | 927 (** 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 | 928 |

adam@354 | 929 Fixpoint map A B (f : A -> computation B) (ls : list A) : computation (list B) := |

adam@354 | 930 match ls with |

adam@354 | 931 | nil => Return nil |

adam@354 | 932 | x :: ls' => Bind (f x) (fun x' => |

adam@354 | 933 Bind (map f ls') (fun ls'' => |

adam@354 | 934 Return (x' :: ls''))) |

adam@354 | 935 end. |

adam@354 | 936 |

adam@355 | 937 (** remove printing exists *) |

adam@356 | 938 Theorem test_map : run (map (fun x => Return (S x)) (1 :: 2 :: 3 :: nil)) |

adam@356 | 939 (2 :: 3 :: 4 :: nil). |

adam@354 | 940 exists 1; reflexivity. |

adam@354 | 941 Qed. |

adam@356 | 942 |

adam@524 | 943 (** 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 | 944 |

adam@357 | 945 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. *) |