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Co-inductive non-termination monads
author Adam Chlipala <adam@chlipala.net>
date Wed, 26 Oct 2011 18:37:47 -0400
parents 3322367e955d
children 62fdf0993e05
rev   line source
adam@350 1 (* Copyright (c) 2006, 2011, 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@353 23 What exactly are the conservative criteria that we run up against? For %\emph{%#<i>#recursive#</i>#%}% definitions, recursive calls are only allowed on %\emph{%#<i>#syntactic subterms#</i>#%}% of the original primary argument, a restriction known as %\index{primitive recursion}\emph{%#<i>#primitive recursion#</i>#%}%. 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 %\emph{%#<i>#co-recursive#</i>#%}% definitions are checked against a syntactic guardness 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@353 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}\emph{%#<i>#deep embedding#</i>#%}% 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}\emph{%#<i>#shallow embedding#</i>#%}%, 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@351 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}\emph{%#<i>#well-founded relation#</i>#%}%, and the associated standard technique in Coq is %\index{well-founded recursion}\emph{%#<i>#well-founded recursion#</i>#%}%. 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@351 37 (** We have a set equipped with some %``%#"#less-than-or-equal-to#"#%''% test. *)
adam@351 38
adam@351 39 (** A standard function inserts an element into a sorted list, preserving sortedness. *)
adam@351 40
adam@351 41 Fixpoint insert (x : A) (ls : list A) : list A :=
adam@351 42 match ls with
adam@351 43 | nil => x :: nil
adam@351 44 | h :: ls' =>
adam@351 45 if le x h
adam@351 46 then x :: ls
adam@351 47 else h :: insert x ls'
adam@351 48 end.
adam@351 49
adam@351 50 (** We will also need a function to merge two sorted lists. (We use a less efficient implementation than usual, because the more efficient implementation already forces us to think about well-founded recursion, while here we are only interested in setting up the example of merge sort.) *)
adam@351 51
adam@351 52 Fixpoint merge (ls1 ls2 : list A) : list A :=
adam@351 53 match ls1 with
adam@351 54 | nil => ls2
adam@351 55 | h :: ls' => insert h (merge ls' ls2)
adam@351 56 end.
adam@351 57
adam@351 58 (** The last helper function for classic merge sort is the one that follows, to partition a list arbitrarily into two pieces of approximately equal length. *)
adam@351 59
adam@351 60 Fixpoint partition (ls : list A) : list A * list A :=
adam@351 61 match ls with
adam@351 62 | nil => (nil, nil)
adam@351 63 | h :: nil => (h :: nil, nil)
adam@351 64 | h1 :: h2 :: ls' =>
adam@351 65 let (ls1, ls2) := partition ls' in
adam@351 66 (h1 :: ls1, h2 :: ls2)
adam@351 67 end.
adam@351 68
adam@351 69 (** Now, let us try to write the final sorting function, using a natural number %``%#"#[<=]#"#%''% test [leb] from the standard library.
adam@351 70 [[
adam@351 71 Fixpoint mergeSort (ls : list A) : list A :=
adam@351 72 if leb (length ls) 2
adam@351 73 then ls
adam@351 74 else let lss := partition ls in
adam@351 75 merge (mergeSort (fst lss)) (mergeSort (snd lss)).
adam@351 76 ]]
adam@351 77
adam@351 78 <<
adam@351 79 Recursive call to mergeSort has principal argument equal to
adam@351 80 "fst (partition ls)" instead of a subterm of "ls".
adam@351 81 >>
adam@351 82
adam@351 83 The definition is rejected for not following the simple primitive recursion criterion. In particular, it is not apparent that recursive calls to [mergeSort] are syntactic subterms of the original argument [ls]; indeed, they are not, yet we know this is a well-founded recursive definition.
adam@351 84
adam@351 85 To produce an acceptable definition, we need to choose a well-founded relation and prove that [mergeSort] respects it. A good starting point is an examination of how well-foundedness is formalized in the Coq standard library. *)
adam@351 86
adam@351 87 Print well_founded.
adam@351 88 (** %\vspace{-.15in}% [[
adam@351 89 well_founded =
adam@351 90 fun (A : Type) (R : A -> A -> Prop) => forall a : A, Acc R a
adam@351 91 ]]
adam@351 92
adam@351 93 The bulk of the definitional work devolves to the %\index{accessibility relation}\index{Gallina terms!Acc}\emph{%#<i>#accessibility#</i>#%}% relation [Acc], whose definition we may also examine. *)
adam@351 94
adam@351 95 Print Acc.
adam@351 96 (** %\vspace{-.15in}% [[
adam@351 97 Inductive Acc (A : Type) (R : A -> A -> Prop) (x : A) : Prop :=
adam@351 98 Acc_intro : (forall y : A, R y x -> Acc R y) -> Acc R x
adam@351 99 ]]
adam@351 100
adam@353 101 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 which 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@351 102
adam@351 103 CoInductive isChain A (R : A -> A -> Prop) : stream A -> Prop :=
adam@351 104 | ChainCons : forall x y s, isChain R (Cons y s)
adam@351 105 -> R y x
adam@351 106 -> isChain R (Cons x (Cons y s)).
adam@351 107
adam@351 108 (** We can now prove that any accessible element cannot be the beginning of any infinite decreasing chain. *)
adam@351 109
adam@351 110 (* begin thide *)
adam@351 111 Lemma noChains' : forall A (R : A -> A -> Prop) x, Acc R x
adam@351 112 -> forall s, ~isChain R (Cons x s).
adam@351 113 induction 1; crush;
adam@351 114 match goal with
adam@351 115 | [ H : isChain _ _ |- _ ] => inversion H; eauto
adam@351 116 end.
adam@351 117 Qed.
adam@351 118
adam@351 119 (** From here, the absence of infinite decreasing chains in well-founded sets is immediate. *)
adam@351 120
adam@351 121 Theorem noChains : forall A (R : A -> A -> Prop), well_founded R
adam@351 122 -> forall s, ~isChain R s.
adam@351 123 destruct s; apply noChains'; auto.
adam@351 124 Qed.
adam@351 125 (* end thide *)
adam@351 126
adam@351 127 (** 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@351 128
adam@351 129 Check Fix.
adam@351 130 (** %\vspace{-.15in}%[[
adam@351 131 Fix
adam@351 132 : forall (A : Type) (R : A -> A -> Prop),
adam@351 133 well_founded R ->
adam@351 134 forall P : A -> Type,
adam@351 135 (forall x : A, (forall y : A, R y x -> P y) -> P x) ->
adam@351 136 forall x : A, P x
adam@351 137 ]]
adam@351 138
adam@351 139 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@351 140
adam@351 141 [[
adam@351 142 forall x : A, (forall y : A, R y x -> P y) -> P x
adam@351 143 ]]
adam@351 144
adam@351 145 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@351 146
adam@353 147 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@351 148
adam@351 149 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@351 150
adam@351 151 Definition lengthOrder (ls1 ls2 : list A) :=
adam@351 152 length ls1 < length ls2.
adam@351 153
adam@353 154 (** 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@351 155
adam@351 156 Hint Constructors Acc.
adam@351 157
adam@351 158 Lemma lengthOrder_wf' : forall len, forall ls, length ls <= len -> Acc lengthOrder ls.
adam@351 159 unfold lengthOrder; induction len; crush.
adam@351 160 Defined.
adam@351 161
adam@351 162 Theorem lengthOrder_wf : well_founded lengthOrder.
adam@351 163 red; intro; eapply lengthOrder_wf'; eauto.
adam@351 164 Defined.
adam@351 165
adam@353 166 (** Notice that we end these proofs with %\index{Vernacular commands!Defined}%[Defined], not [Qed]. Recall that [Defined] marks the theorems as %\emph{transparent}%, so that the details of their proofs may be used during program execution. Why could such details possibly matter for computation? It turns out that [Fix] satisfies the primitive recursion restriction by declaring itself as %\emph{%#<i>#recursive in the structure of [Acc] proofs#</i>#%}%. 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@351 167
adam@351 168 To justify our two recursive [mergeSort] calls, we will also need to prove that [partition] respects the [lengthOrder] relation. These proofs, too, must be kept transparent, to avoid stuckness of [Fix] evaluation. *)
adam@351 169
adam@351 170 Lemma partition_wf : forall len ls, 2 <= length ls <= len
adam@351 171 -> let (ls1, ls2) := partition ls in
adam@351 172 lengthOrder ls1 ls /\ lengthOrder ls2 ls.
adam@351 173 unfold lengthOrder; induction len; crush; do 2 (destruct ls; crush);
adam@351 174 destruct (le_lt_dec 2 (length ls));
adam@351 175 repeat (match goal with
adam@351 176 | [ _ : length ?E < 2 |- _ ] => destruct E
adam@351 177 | [ _ : S (length ?E) < 2 |- _ ] => destruct E
adam@351 178 | [ IH : _ |- context[partition ?L] ] =>
adam@351 179 specialize (IH L); destruct (partition L); destruct IH
adam@351 180 end; crush).
adam@351 181 Defined.
adam@351 182
adam@351 183 Ltac partition := intros ls ?; intros; generalize (@partition_wf (length ls) ls);
adam@351 184 destruct (partition ls); destruct 1; crush.
adam@351 185
adam@351 186 Lemma partition_wf1 : forall ls, 2 <= length ls
adam@351 187 -> lengthOrder (fst (partition ls)) ls.
adam@351 188 partition.
adam@351 189 Defined.
adam@351 190
adam@351 191 Lemma partition_wf2 : forall ls, 2 <= length ls
adam@351 192 -> lengthOrder (snd (partition ls)) ls.
adam@351 193 partition.
adam@351 194 Defined.
adam@351 195
adam@351 196 Hint Resolve partition_wf1 partition_wf2.
adam@351 197
adam@353 198 (** 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. *)
adam@351 199
adam@351 200 Definition mergeSort : list A -> list A.
adam@351 201 (* begin thide *)
adam@351 202 refine (Fix lengthOrder_wf (fun _ => list A)
adam@351 203 (fun (ls : list A)
adam@351 204 (mergeSort : forall ls' : list A, lengthOrder ls' ls -> list A) =>
adam@351 205 if le_lt_dec 2 (length ls)
adam@351 206 then let lss := partition ls in
adam@351 207 merge (mergeSort (fst lss) _) (mergeSort (snd lss) _)
adam@351 208 else ls)); subst lss; eauto.
adam@351 209 Defined.
adam@351 210 (* end thide *)
adam@351 211 End mergeSort.
adam@351 212
adam@351 213 (** The important thing is that it is now easy to evaluate calls to [mergeSort]. *)
adam@351 214
adam@351 215 Eval compute in mergeSort leb (1 :: 2 :: 36 :: 8 :: 19 :: nil).
adam@351 216 (** [= 1 :: 2 :: 8 :: 19 :: 36 :: nil] *)
adam@351 217
adam@351 218 (** 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@351 219
adam@351 220 (* begin thide *)
adam@351 221 Theorem mergeSort_eq : forall A (le : A -> A -> bool) ls,
adam@351 222 mergeSort le ls = if le_lt_dec 2 (length ls)
adam@351 223 then let lss := partition ls in
adam@351 224 merge le (mergeSort le (fst lss)) (mergeSort le (snd lss))
adam@351 225 else ls.
adam@351 226 intros; apply (Fix_eq (@lengthOrder_wf A) (fun _ => list A)); intros.
adam@351 227
adam@351 228 (** 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}\emph{%#<i>#function extensionality#</i>#%}% property is neither provable nor disprovable within Coq. The type of [Fix_eq] makes clear what we must show manually: *)
adam@351 229
adam@351 230 Check Fix_eq.
adam@351 231 (** %\vspace{-.15in}%[[
adam@351 232 Fix_eq
adam@351 233 : forall (A : Type) (R : A -> A -> Prop) (Rwf : well_founded R)
adam@351 234 (P : A -> Type)
adam@351 235 (F : forall x : A, (forall y : A, R y x -> P y) -> P x),
adam@351 236 (forall (x : A) (f g : forall y : A, R y x -> P y),
adam@351 237 (forall (y : A) (p : R y x), f y p = g y p) -> F x f = F x g) ->
adam@351 238 forall x : A,
adam@351 239 Fix Rwf P F x = F x (fun (y : A) (_ : R y x) => Fix Rwf P F y)
adam@351 240 ]]
adam@351 241
adam@351 242 Most such obligations are dischargable with straightforward proof automation, and this example is no exception. *)
adam@351 243
adam@351 244 match goal with
adam@351 245 | [ |- context[match ?E with left _ => _ | right _ => _ end] ] => destruct E
adam@351 246 end; simpl; f_equal; auto.
adam@351 247 Qed.
adam@351 248 (* end thide *)
adam@351 249
adam@351 250 (** As a final test of our definition's suitability, we can extract to OCaml. *)
adam@351 251
adam@351 252 Extraction mergeSort.
adam@351 253
adam@351 254 (** <<
adam@351 255 let rec mergeSort le x =
adam@351 256 match le_lt_dec (S (S O)) (length x) with
adam@351 257 | Left ->
adam@351 258 let lss = partition x in
adam@351 259 merge le (mergeSort le (fst lss)) (mergeSort le (snd lss))
adam@351 260 | Right -> x
adam@351 261 >>
adam@351 262
adam@353 263 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@351 264
adam@351 265 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@351 266
adam@351 267 Check well_founded_induction.
adam@351 268 (** %\vspace{-.15in}%[[
adam@351 269 well_founded_induction
adam@351 270 : forall (A : Type) (R : A -> A -> Prop),
adam@351 271 well_founded R ->
adam@351 272 forall P : A -> Set,
adam@351 273 (forall x : A, (forall y : A, R y x -> P y) -> P x) ->
adam@351 274 forall a : A, P a
adam@351 275 ]]
adam@351 276
adam@351 277 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@352 278
adam@352 279
adam@354 280 (** * A Non-Termination Monad Inspired by Domain Theory *)
adam@352 281
adam@352 282 Section computation.
adam@352 283 Variable A : Type.
adam@352 284
adam@352 285 Definition computation :=
adam@352 286 {f : nat -> option A
adam@352 287 | forall (n : nat) (v : A),
adam@352 288 f n = Some v
adam@352 289 -> forall (n' : nat), n' >= n
adam@352 290 -> f n' = Some v}.
adam@352 291
adam@352 292 Definition runTo (m : computation) (n : nat) (v : A) :=
adam@352 293 proj1_sig m n = Some v.
adam@352 294
adam@352 295 Definition run (m : computation) (v : A) :=
adam@352 296 exists n, runTo m n v.
adam@352 297 End computation.
adam@352 298
adam@352 299 Hint Unfold runTo.
adam@352 300
adam@352 301 Ltac run' := unfold run, runTo in *; try red; crush;
adam@352 302 repeat (match goal with
adam@352 303 | [ _ : proj1_sig ?E _ = _ |- _ ] =>
adam@352 304 match goal with
adam@352 305 | [ x : _ |- _ ] =>
adam@352 306 match x with
adam@352 307 | E => destruct E
adam@352 308 end
adam@352 309 end
adam@352 310 | [ |- context[match ?M with exist _ _ => _ end] ] => let Heq := fresh "Heq" in
adam@352 311 case_eq M; intros ? ? Heq; try rewrite Heq in *; try subst
adam@352 312 | [ _ : context[match ?M with exist _ _ => _ end] |- _ ] => let Heq := fresh "Heq" in
adam@352 313 case_eq M; intros ? ? Heq; try rewrite Heq in *; subst
adam@352 314 | [ H : forall n v, ?E n = Some v -> _,
adam@352 315 _ : context[match ?E ?N with Some _ => _ | None => _ end] |- _ ] =>
adam@352 316 specialize (H N); destruct (E N); try rewrite (H _ (refl_equal _)) by auto; try discriminate
adam@352 317 | [ H : forall n v, ?E n = Some v -> _, H' : ?E _ = Some _ |- _ ] => rewrite (H _ _ H') by auto
adam@352 318 end; simpl in *); eauto 7.
adam@352 319
adam@352 320 Ltac run := run'; repeat (match goal with
adam@352 321 | [ H : forall n v, ?E n = Some v -> _
adam@352 322 |- context[match ?E ?N with Some _ => _ | None => _ end] ] =>
adam@352 323 specialize (H N); destruct (E N); try rewrite (H _ (refl_equal _)) by auto; try discriminate
adam@352 324 end; run').
adam@352 325
adam@352 326 Lemma ex_irrelevant : forall P : Prop, P -> exists n : nat, P.
adam@352 327 exists 0; auto.
adam@352 328 Qed.
adam@352 329
adam@352 330 Hint Resolve ex_irrelevant.
adam@352 331
adam@352 332 Require Import Max.
adam@352 333
adam@352 334 Ltac max := intros n m; generalize (max_spec_le n m); crush.
adam@352 335
adam@352 336 Lemma max_1 : forall n m, max n m >= n.
adam@352 337 max.
adam@352 338 Qed.
adam@352 339
adam@352 340 Lemma max_2 : forall n m, max n m >= m.
adam@352 341 max.
adam@352 342 Qed.
adam@352 343
adam@352 344 Hint Resolve max_1 max_2.
adam@352 345
adam@352 346 Lemma ge_refl : forall n, n >= n.
adam@352 347 crush.
adam@352 348 Qed.
adam@352 349
adam@352 350 Hint Resolve ge_refl.
adam@352 351
adam@352 352 Hint Extern 1 => match goal with
adam@352 353 | [ H : _ = exist _ _ _ |- _ ] => rewrite H
adam@352 354 end.
adam@352 355
adam@352 356 Section Bottom.
adam@352 357 Variable A : Type.
adam@352 358
adam@352 359 Definition Bottom : computation A.
adam@352 360 exists (fun _ : nat => @None A); abstract run.
adam@352 361 Defined.
adam@352 362
adam@352 363 Theorem run_Bottom : forall v, ~run Bottom v.
adam@352 364 run.
adam@352 365 Qed.
adam@352 366 End Bottom.
adam@352 367
adam@352 368 Section Return.
adam@352 369 Variable A : Type.
adam@352 370 Variable v : A.
adam@352 371
adam@352 372 Definition Return : computation A.
adam@352 373 intros; exists (fun _ : nat => Some v); abstract run.
adam@352 374 Defined.
adam@352 375
adam@352 376 Theorem run_Return : run Return v.
adam@352 377 run.
adam@352 378 Qed.
adam@352 379
adam@352 380 Theorem run_Return_inv : forall x, run Return x -> x = v.
adam@352 381 run.
adam@352 382 Qed.
adam@352 383 End Return.
adam@352 384
adam@352 385 Hint Resolve run_Return.
adam@352 386
adam@352 387 Section Bind.
adam@352 388 Variables A B : Type.
adam@352 389 Variable m1 : computation A.
adam@352 390 Variable m2 : A -> computation B.
adam@352 391
adam@352 392 Definition Bind : computation B.
adam@352 393 exists (fun n =>
adam@352 394 let (f1, Hf1) := m1 in
adam@352 395 match f1 n with
adam@352 396 | None => None
adam@352 397 | Some v =>
adam@352 398 let (f2, Hf2) := m2 v in
adam@352 399 f2 n
adam@352 400 end); abstract run.
adam@352 401 Defined.
adam@352 402
adam@352 403 Require Import Max.
adam@352 404
adam@352 405 Theorem run_Bind : forall (v1 : A) (v2 : B),
adam@352 406 run m1 v1
adam@352 407 -> run (m2 v1) v2
adam@352 408 -> run Bind v2.
adam@352 409 run; match goal with
adam@352 410 | [ x : nat, y : nat |- _ ] => exists (max x y)
adam@352 411 end; run.
adam@352 412 Qed.
adam@352 413
adam@352 414 Theorem run_Bind_inv : forall (v2 : B),
adam@352 415 run Bind v2
adam@352 416 -> exists v1 : A,
adam@352 417 run m1 v1
adam@352 418 /\ run (m2 v1) v2.
adam@352 419 run.
adam@352 420 Qed.
adam@352 421 End Bind.
adam@352 422
adam@352 423 Hint Resolve run_Bind.
adam@352 424
adam@352 425 Notation "x <- m1 ; m2" :=
adam@352 426 (Bind m1 (fun x => m2)) (right associativity, at level 70).
adam@352 427
adam@352 428 Definition meq A (m1 m2 : computation A) := forall n, proj1_sig m1 n = proj1_sig m2 n.
adam@352 429
adam@352 430 Theorem left_identity : forall A B (a : A) (f : A -> computation B),
adam@352 431 meq (Bind (Return a) f) (f a).
adam@352 432 run.
adam@352 433 Qed.
adam@352 434
adam@352 435 Theorem right_identity : forall A (m : computation A),
adam@352 436 meq (Bind m (@Return _)) m.
adam@352 437 run.
adam@352 438 Qed.
adam@352 439
adam@352 440 Theorem associativity : forall A B C (m : computation A) (f : A -> computation B) (g : B -> computation C),
adam@352 441 meq (Bind (Bind m f) g) (Bind m (fun x => Bind (f x) g)).
adam@352 442 run.
adam@352 443 Qed.
adam@352 444
adam@352 445 Section monotone_runTo.
adam@352 446 Variable A : Type.
adam@352 447 Variable c : computation A.
adam@352 448 Variable v : A.
adam@352 449
adam@352 450 Theorem monotone_runTo : forall (n1 : nat),
adam@352 451 runTo c n1 v
adam@352 452 -> forall n2, n2 >= n1
adam@352 453 -> runTo c n2 v.
adam@352 454 run.
adam@352 455 Qed.
adam@352 456 End monotone_runTo.
adam@352 457
adam@352 458 Hint Resolve monotone_runTo.
adam@352 459
adam@352 460 Section lattice.
adam@352 461 Variable A : Type.
adam@352 462
adam@352 463 Definition leq (x y : option A) :=
adam@352 464 forall v, x = Some v -> y = Some v.
adam@352 465 End lattice.
adam@352 466
adam@352 467 Hint Unfold leq.
adam@352 468
adam@352 469 Section Fix.
adam@352 470 Variables A B : Type.
adam@352 471 Variable f : (A -> computation B) -> (A -> computation B).
adam@352 472
adam@352 473 Hypothesis f_continuous : forall n v v1 x,
adam@352 474 runTo (f v1 x) n v
adam@352 475 -> forall (v2 : A -> computation B),
adam@352 476 (forall x, leq (proj1_sig (v1 x) n) (proj1_sig (v2 x) n))
adam@352 477 -> runTo (f v2 x) n v.
adam@352 478
adam@352 479 Fixpoint Fix' (n : nat) (x : A) : computation B :=
adam@352 480 match n with
adam@352 481 | O => Bottom _
adam@352 482 | S n' => f (Fix' n') x
adam@352 483 end.
adam@352 484
adam@352 485 Hint Extern 1 (_ >= _) => omega.
adam@352 486 Hint Unfold leq.
adam@352 487
adam@352 488 Lemma Fix'_ok : forall steps n x v, proj1_sig (Fix' n x) steps = Some v
adam@352 489 -> forall n', n' >= n
adam@352 490 -> proj1_sig (Fix' n' x) steps = Some v.
adam@352 491 unfold runTo in *; induction n; crush;
adam@352 492 match goal with
adam@352 493 | [ H : _ >= _ |- _ ] => inversion H; crush; eauto
adam@352 494 end.
adam@352 495 Qed.
adam@352 496
adam@352 497 Hint Resolve Fix'_ok.
adam@352 498
adam@352 499 Hint Extern 1 (proj1_sig _ _ = _) => simpl;
adam@352 500 match goal with
adam@352 501 | [ |- proj1_sig ?E _ = _ ] => eapply (proj2_sig E)
adam@352 502 end.
adam@352 503
adam@352 504 Definition Fix : A -> computation B.
adam@352 505 intro x; exists (fun n => proj1_sig (Fix' n x) n); abstract run.
adam@352 506 Defined.
adam@352 507
adam@352 508 Definition extensional (f : (A -> computation B) -> (A -> computation B)) :=
adam@352 509 forall g1 g2 n,
adam@352 510 (forall x, proj1_sig (g1 x) n = proj1_sig (g2 x) n)
adam@352 511 -> forall x, proj1_sig (f g1 x) n = proj1_sig (f g2 x) n.
adam@352 512
adam@352 513 Hypothesis f_extensional : extensional f.
adam@352 514
adam@352 515 Theorem run_Fix : forall x v,
adam@352 516 run (f Fix x) v
adam@352 517 -> run (Fix x) v.
adam@352 518 run; match goal with
adam@352 519 | [ n : nat |- _ ] => exists (S n); eauto
adam@352 520 end.
adam@352 521 Qed.
adam@352 522 End Fix.
adam@352 523
adam@352 524 Hint Resolve run_Fix.
adam@352 525
adam@352 526 Lemma leq_Some : forall A (x y : A), leq (Some x) (Some y)
adam@352 527 -> x = y.
adam@352 528 intros ? ? ? H; generalize (H _ (refl_equal _)); crush.
adam@352 529 Qed.
adam@352 530
adam@352 531 Lemma leq_None : forall A (x y : A), leq (Some x) None
adam@352 532 -> False.
adam@352 533 intros ? ? ? H; generalize (H _ (refl_equal _)); crush.
adam@352 534 Qed.
adam@352 535
adam@352 536 Definition mergeSort' : forall A, (A -> A -> bool) -> list A -> computation (list A).
adam@352 537 refine (fun A le => Fix
adam@352 538 (fun (mergeSort : list A -> computation (list A))
adam@352 539 (ls : list A) =>
adam@352 540 if le_lt_dec 2 (length ls)
adam@352 541 then let lss := partition ls in
adam@352 542 ls1 <- mergeSort (fst lss);
adam@352 543 ls2 <- mergeSort (snd lss);
adam@352 544 Return (merge le ls1 ls2)
adam@352 545 else Return ls) _); abstract (run;
adam@352 546 repeat (match goal with
adam@352 547 | [ |- context[match ?E with O => _ | S _ => _ end] ] => destruct E
adam@352 548 end; run);
adam@352 549 repeat match goal with
adam@352 550 | [ H : forall x, leq (proj1_sig (?f x) _) (proj1_sig (?g x) _) |- _ ] =>
adam@352 551 match goal with
adam@352 552 | [ H1 : f ?arg = _, H2 : g ?arg = _ |- _ ] =>
adam@352 553 generalize (H arg); rewrite H1; rewrite H2; clear H1 H2; simpl; intro
adam@352 554 end
adam@352 555 end; run; repeat match goal with
adam@352 556 | [ H : _ |- _ ] => (apply leq_None in H; tauto) || (apply leq_Some in H; subst)
adam@352 557 end; auto).
adam@352 558 Defined.
adam@352 559
adam@352 560 Print mergeSort'.
adam@352 561
adam@352 562 Lemma test_mergeSort' : run (mergeSort' leb (1 :: 2 :: 36 :: 8 :: 19 :: nil))
adam@352 563 (1 :: 2 :: 8 :: 19 :: 36 :: nil).
adam@352 564 exists 4; reflexivity.
adam@352 565 Qed.
adam@352 566
adam@352 567 Definition looper : bool -> computation unit.
adam@352 568 refine (Fix (fun looper (b : bool) =>
adam@352 569 if b then Return tt else looper b) _);
adam@352 570 abstract (unfold leq in *; run;
adam@352 571 repeat match goal with
adam@352 572 | [ x : unit |- _ ] => destruct x
adam@352 573 | [ x : bool |- _ ] => destruct x
adam@352 574 end; auto).
adam@352 575 Defined.
adam@352 576
adam@352 577 Lemma test_looper : run (looper true) tt.
adam@352 578 exists 1; reflexivity.
adam@352 579 Qed.
adam@354 580
adam@354 581
adam@354 582 (** * Co-Inductive Non-Termination Monads *)
adam@354 583
adam@354 584 CoInductive thunk (A : Type) : Type :=
adam@354 585 | Answer : A -> thunk A
adam@354 586 | Think : thunk A -> thunk A.
adam@354 587
adam@354 588 CoFixpoint TBind (A B : Type) (m1 : thunk A) (m2 : A -> thunk B) : thunk B :=
adam@354 589 match m1 with
adam@354 590 | Answer x => m2 x
adam@354 591 | Think m1' => Think (TBind m1' m2)
adam@354 592 end.
adam@354 593
adam@354 594 CoInductive thunk_eq A : thunk A -> thunk A -> Prop :=
adam@354 595 | EqAnswer : forall x, thunk_eq (Answer x) (Answer x)
adam@354 596 | EqThinkL : forall m1 m2, thunk_eq m1 m2 -> thunk_eq (Think m1) m2
adam@354 597 | EqThinkR : forall m1 m2, thunk_eq m1 m2 -> thunk_eq m1 (Think m2).
adam@354 598
adam@354 599 Section thunk_eq_coind.
adam@354 600 Variable A : Type.
adam@354 601 Variable P : thunk A -> thunk A -> Prop.
adam@354 602
adam@354 603 Hypothesis H : forall m1 m2, P m1 m2
adam@354 604 -> match m1, m2 with
adam@354 605 | Answer x1, Answer x2 => x1 = x2
adam@354 606 | Think m1', Think m2' => P m1' m2'
adam@354 607 | Think m1', _ => P m1' m2
adam@354 608 | _, Think m2' => P m1 m2'
adam@354 609 end.
adam@354 610
adam@354 611 Theorem thunk_eq_coind : forall m1 m2, P m1 m2 -> thunk_eq m1 m2.
adam@354 612 cofix; intros;
adam@354 613 match goal with
adam@354 614 | [ H' : P _ _ |- _ ] => specialize (H H'); clear H'
adam@354 615 end; destruct m1; destruct m2; subst; repeat constructor; auto.
adam@354 616 Qed.
adam@354 617 End thunk_eq_coind.
adam@354 618
adam@354 619 Definition frob A (m : thunk A) : thunk A :=
adam@354 620 match m with
adam@354 621 | Answer x => Answer x
adam@354 622 | Think m' => Think m'
adam@354 623 end.
adam@354 624
adam@354 625 Theorem frob_eq : forall A (m : thunk A), frob m = m.
adam@354 626 destruct m; reflexivity.
adam@354 627 Qed.
adam@354 628
adam@354 629 Theorem thunk_eq_frob : forall A (m1 m2 : thunk A),
adam@354 630 thunk_eq (frob m1) (frob m2)
adam@354 631 -> thunk_eq m1 m2.
adam@354 632 intros; repeat rewrite frob_eq in *; auto.
adam@354 633 Qed.
adam@354 634
adam@354 635 Ltac findDestr := match goal with
adam@354 636 | [ |- context[match ?E with Answer _ => _ | Think _ => _ end] ] =>
adam@354 637 match E with
adam@354 638 | context[match _ with Answer _ => _ | Think _ => _ end] => fail 1
adam@354 639 | _ => destruct E
adam@354 640 end
adam@354 641 end.
adam@354 642
adam@354 643 Theorem thunk_eq_refl : forall A (m : thunk A), thunk_eq m m.
adam@354 644 intros; apply (thunk_eq_coind (fun m1 m2 => m1 = m2)); crush; findDestr; reflexivity.
adam@354 645 Qed.
adam@354 646
adam@354 647 Hint Resolve thunk_eq_refl.
adam@354 648
adam@354 649 Theorem tleft_identity : forall A B (a : A) (f : A -> thunk B),
adam@354 650 thunk_eq (TBind (Answer a) f) (f a).
adam@354 651 intros; apply thunk_eq_frob; crush.
adam@354 652 Qed.
adam@354 653
adam@354 654 Theorem tright_identity : forall A (m : thunk A),
adam@354 655 thunk_eq (TBind m (@Answer _)) m.
adam@354 656 intros; apply (thunk_eq_coind (fun m1 m2 => m1 = TBind m2 (@Answer _))); crush;
adam@354 657 findDestr; reflexivity.
adam@354 658 Qed.
adam@354 659
adam@354 660 Lemma TBind_Answer : forall (A B : Type) (v : A) (m2 : A -> thunk B),
adam@354 661 TBind (Answer v) m2 = m2 v.
adam@354 662 intros; rewrite <- (frob_eq (TBind (Answer v) m2));
adam@354 663 simpl; findDestr; reflexivity.
adam@354 664 Qed.
adam@354 665
adam@354 666 Hint Rewrite TBind_Answer : cpdt.
adam@354 667
adam@354 668 Theorem tassociativity : forall A B C (m : thunk A) (f : A -> thunk B) (g : B -> thunk C),
adam@354 669 thunk_eq (TBind (TBind m f) g) (TBind m (fun x => TBind (f x) g)).
adam@354 670 intros; apply (thunk_eq_coind (fun m1 m2 => (exists m,
adam@354 671 m1 = TBind (TBind m f) g
adam@354 672 /\ m2 = TBind m (fun x => TBind (f x) g))
adam@354 673 \/ m1 = m2)); crush; eauto; repeat (findDestr; crush; eauto).
adam@354 674 Qed.
adam@354 675
adam@354 676 CoFixpoint fact (n acc : nat) : thunk nat :=
adam@354 677 match n with
adam@354 678 | O => Answer acc
adam@354 679 | S n' => Think (fact n' (S n' * acc))
adam@354 680 end.
adam@354 681
adam@354 682 Inductive eval A : thunk A -> A -> Prop :=
adam@354 683 | EvalAnswer : forall x, eval (Answer x) x
adam@354 684 | EvalThink : forall m x, eval m x -> eval (Think m) x.
adam@354 685
adam@354 686 Hint Rewrite frob_eq : cpdt.
adam@354 687
adam@354 688 Lemma eval_frob : forall A (c : thunk A) x,
adam@354 689 eval (frob c) x
adam@354 690 -> eval c x.
adam@354 691 crush.
adam@354 692 Qed.
adam@354 693
adam@354 694 Theorem eval_fact : eval (fact 5 1) 120.
adam@354 695 repeat (apply eval_frob; simpl; constructor).
adam@354 696 Qed.
adam@354 697
adam@354 698 (** [[
adam@354 699 CoFixpoint fib (n : nat) : thunk nat :=
adam@354 700 match n with
adam@354 701 | 0 => Answer 1
adam@354 702 | 1 => Answer 1
adam@354 703 | _ => TBind (fib (pred n)) (fun n1 =>
adam@354 704 TBind (fib (pred (pred n))) (fun n2 =>
adam@354 705 Answer (n1 + n2)))
adam@354 706 end.
adam@354 707 ]]
adam@354 708 *)
adam@354 709
adam@354 710
adam@354 711 CoInductive comp (A : Type) : Type :=
adam@354 712 | Ret : A -> comp A
adam@354 713 | Bnd : forall B, comp B -> (B -> comp A) -> comp A.
adam@354 714
adam@354 715 Inductive exec A : comp A -> A -> Prop :=
adam@354 716 | ExecRet : forall x, exec (Ret x) x
adam@354 717 | ExecBnd : forall B (c : comp B) (f : B -> comp A) x1 x2, exec (A := B) c x1
adam@354 718 -> exec (f x1) x2
adam@354 719 -> exec (Bnd c f) x2.
adam@354 720
adam@354 721 Hint Constructors exec.
adam@354 722
adam@354 723 Definition comp_eq A (c1 c2 : comp A) := forall r, exec c1 r <-> exec c2 r.
adam@354 724
adam@354 725 Ltac inverter := repeat match goal with
adam@354 726 | [ H : exec _ _ |- _ ] => inversion H; []; crush
adam@354 727 end.
adam@354 728
adam@354 729 Theorem cleft_identity : forall A B (a : A) (f : A -> comp B),
adam@354 730 comp_eq (Bnd (Ret a) f) (f a).
adam@354 731 red; crush; inverter; eauto.
adam@354 732 Qed.
adam@354 733
adam@354 734 Theorem cright_identity : forall A (m : comp A),
adam@354 735 comp_eq (Bnd m (@Ret _)) m.
adam@354 736 red; crush; inverter; eauto.
adam@354 737 Qed.
adam@354 738
adam@354 739 Lemma cassociativity1 : forall A B C (f : A -> comp B) (g : B -> comp C) r c,
adam@354 740 exec c r
adam@354 741 -> forall m, c = Bnd (Bnd m f) g
adam@354 742 -> exec (Bnd m (fun x => Bnd (f x) g)) r.
adam@354 743 induction 1; crush.
adam@354 744 match goal with
adam@354 745 | [ H : Bnd _ _ = Bnd _ _ |- _ ] => injection H; clear H; intros; try subst
adam@354 746 end.
adam@354 747 move H3 after A.
adam@354 748 generalize dependent B0.
adam@354 749 do 2 intro.
adam@354 750 subst.
adam@354 751 crush.
adam@354 752 inversion H; clear H; crush.
adam@354 753 eauto.
adam@354 754 Qed.
adam@354 755
adam@354 756 Lemma cassociativity2 : forall A B C (f : A -> comp B) (g : B -> comp C) r c,
adam@354 757 exec c r
adam@354 758 -> forall m, c = Bnd m (fun x => Bnd (f x) g)
adam@354 759 -> exec (Bnd (Bnd m f) g) r.
adam@354 760 induction 1; crush.
adam@354 761 match goal with
adam@354 762 | [ H : Bnd _ _ = Bnd _ _ |- _ ] => injection H; clear H; intros; try subst
adam@354 763 end.
adam@354 764 move H3 after B.
adam@354 765 generalize dependent B0.
adam@354 766 do 2 intro.
adam@354 767 subst.
adam@354 768 crush.
adam@354 769 inversion H0; clear H0; crush.
adam@354 770 eauto.
adam@354 771 Qed.
adam@354 772
adam@354 773 Hint Resolve cassociativity1 cassociativity2.
adam@354 774
adam@354 775 Theorem cassociativity : forall A B C (m : comp A) (f : A -> comp B) (g : B -> comp C),
adam@354 776 comp_eq (Bnd (Bnd m f) g) (Bnd m (fun x => Bnd (f x) g)).
adam@354 777 red; crush; eauto.
adam@354 778 Qed.
adam@354 779
adam@354 780 CoFixpoint mergeSort'' A (le : A -> A -> bool) (ls : list A) : comp (list A) :=
adam@354 781 if le_lt_dec 2 (length ls)
adam@354 782 then let lss := partition ls in
adam@354 783 Bnd (mergeSort'' le (fst lss)) (fun ls1 =>
adam@354 784 Bnd (mergeSort'' le (snd lss)) (fun ls2 =>
adam@354 785 Ret (merge le ls1 ls2)))
adam@354 786 else Ret ls.
adam@354 787
adam@354 788 Definition frob' A (c : comp A) :=
adam@354 789 match c with
adam@354 790 | Ret x => Ret x
adam@354 791 | Bnd _ c' f => Bnd c' f
adam@354 792 end.
adam@354 793
adam@354 794 Lemma frob'_eq : forall A (c : comp A), frob' c = c.
adam@354 795 destruct c; reflexivity.
adam@354 796 Qed.
adam@354 797
adam@354 798 Hint Rewrite frob'_eq : cpdt.
adam@354 799
adam@354 800 Lemma exec_frob : forall A (c : comp A) x,
adam@354 801 exec (frob' c) x
adam@354 802 -> exec c x.
adam@354 803 crush.
adam@354 804 Qed.
adam@354 805
adam@354 806 Lemma test_mergeSort'' : exec (mergeSort'' leb (1 :: 2 :: 36 :: 8 :: 19 :: nil))
adam@354 807 (1 :: 2 :: 8 :: 19 :: 36 :: nil).
adam@354 808 repeat (apply exec_frob; simpl; econstructor).
adam@354 809 Qed.
adam@354 810
adam@354 811 Definition curriedAdd (n : nat) := Ret (fun m : nat => Ret (n + m)).
adam@354 812
adam@354 813 (** [[
adam@354 814 Definition testCurriedAdd :=
adam@354 815 Bnd (curriedAdd 2) (fun f => f 3).
adam@354 816 ]]
adam@354 817
adam@354 818 <<
adam@354 819 Error: Universe inconsistency.
adam@354 820 >>
adam@354 821 *)
adam@354 822
adam@354 823
adam@354 824 (** * Comparing the Options *)
adam@354 825
adam@354 826 Definition curriedAdd' (n : nat) := Return (fun m : nat => Return (n + m)).
adam@354 827
adam@354 828 Definition testCurriedAdd :=
adam@354 829 Bind (curriedAdd' 2) (fun f => f 3).
adam@354 830
adam@354 831 Fixpoint map A B (f : A -> computation B) (ls : list A) : computation (list B) :=
adam@354 832 match ls with
adam@354 833 | nil => Return nil
adam@354 834 | x :: ls' => Bind (f x) (fun x' =>
adam@354 835 Bind (map f ls') (fun ls'' =>
adam@354 836 Return (x' :: ls'')))
adam@354 837 end.
adam@354 838
adam@354 839 Theorem test_map : run (map (fun x => Return (S x)) (1 :: 2 :: 3 :: nil)) (2 :: 3 :: 4 :: nil).
adam@354 840 exists 1; reflexivity.
adam@354 841 Qed.