author bulwahn Thu Dec 10 11:58:26 2009 +0100 (2009-12-10) changeset 34051 1a82e2e29d67 parent 32069 6d28bbd33e2c child 35113 1a0c129bb2e0 permissions -rw-r--r--
2     Author:     John Matthews, Galois Connections; Alexander Krauss, Lukas Bulwahn & Florian Haftmann, TU Muenchen
3 *)
8 imports Heap
9 begin
11 subsection {* The monad *}
13 subsubsection {* Monad combinators *}
15 datatype exception = Exn
17 text {* Monadic heap actions either produce values
18   and transform the heap, or fail *}
19 datatype 'a Heap = Heap "heap \<Rightarrow> ('a + exception) \<times> heap"
21 primrec
22   execute :: "'a Heap \<Rightarrow> heap \<Rightarrow> ('a + exception) \<times> heap" where
23   "execute (Heap f) = f"
24 lemmas [code del] = execute.simps
26 lemma Heap_execute [simp]:
27   "Heap (execute f) = f" by (cases f) simp_all
29 lemma Heap_eqI:
30   "(\<And>h. execute f h = execute g h) \<Longrightarrow> f = g"
31     by (cases f, cases g) (auto simp: expand_fun_eq)
33 lemma Heap_eqI':
34   "(\<And>h. (\<lambda>x. execute (f x) h) = (\<lambda>y. execute (g y) h)) \<Longrightarrow> f = g"
35     by (auto simp: expand_fun_eq intro: Heap_eqI)
37 lemma Heap_strip: "(\<And>f. PROP P f) \<equiv> (\<And>g. PROP P (Heap g))"
38 proof
39   fix g :: "heap \<Rightarrow> ('a + exception) \<times> heap"
40   assume "\<And>f. PROP P f"
41   then show "PROP P (Heap g)" .
42 next
43   fix f :: "'a Heap"
44   assume assm: "\<And>g. PROP P (Heap g)"
45   then have "PROP P (Heap (execute f))" .
46   then show "PROP P f" by simp
47 qed
49 definition
50   heap :: "(heap \<Rightarrow> 'a \<times> heap) \<Rightarrow> 'a Heap" where
51   [code del]: "heap f = Heap (\<lambda>h. apfst Inl (f h))"
53 lemma execute_heap [simp]:
54   "execute (heap f) h = apfst Inl (f h)"
57 definition
58   bindM :: "'a Heap \<Rightarrow> ('a \<Rightarrow> 'b Heap) \<Rightarrow> 'b Heap" (infixl ">>=" 54) where
59   [code del]: "f >>= g = Heap (\<lambda>h. case execute f h of
60                   (Inl x, h') \<Rightarrow> execute (g x) h'
61                 | r \<Rightarrow> r)"
63 notation
64   bindM (infixl "\<guillemotright>=" 54)
66 abbreviation
67   chainM :: "'a Heap \<Rightarrow> 'b Heap \<Rightarrow> 'b Heap"  (infixl ">>" 54) where
68   "f >> g \<equiv> f >>= (\<lambda>_. g)"
70 notation
71   chainM (infixl "\<guillemotright>" 54)
73 definition
74   return :: "'a \<Rightarrow> 'a Heap" where
75   [code del]: "return x = heap (Pair x)"
77 lemma execute_return [simp]:
78   "execute (return x) h = apfst Inl (x, h)"
81 definition
82   raise :: "string \<Rightarrow> 'a Heap" where -- {* the string is just decoration *}
83   [code del]: "raise s = Heap (Pair (Inr Exn))"
85 notation (latex output)
86   "raise" ("\<^raw:{\textsf{raise}}>")
88 lemma execute_raise [simp]:
89   "execute (raise s) h = (Inr Exn, h)"
93 subsubsection {* do-syntax *}
95 text {*
96   We provide a convenient do-notation for monadic expressions
97   well-known from Haskell.  @{const Let} is printed
98   specially in do-expressions.
99 *}
101 nonterminals do_expr
103 syntax
104   "_do" :: "do_expr \<Rightarrow> 'a"
105     ("(do (_)//done)" [12] 100)
106   "_bindM" :: "pttrn \<Rightarrow> 'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
107     ("_ <- _;//_" [1000, 13, 12] 12)
108   "_chainM" :: "'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
109     ("_;//_" [13, 12] 12)
110   "_let" :: "pttrn \<Rightarrow> 'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
111     ("let _ = _;//_" [1000, 13, 12] 12)
112   "_nil" :: "'a \<Rightarrow> do_expr"
113     ("_" [12] 12)
115 syntax (xsymbols)
116   "_bindM" :: "pttrn \<Rightarrow> 'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
117     ("_ \<leftarrow> _;//_" [1000, 13, 12] 12)
118 syntax (latex output)
119   "_do" :: "do_expr \<Rightarrow> 'a"
120     ("(\<^raw:{\textsf{do}}> (_))" [12] 100)
121   "_let" :: "pttrn \<Rightarrow> 'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
122     ("\<^raw:\textsf{let}> _ = _;//_" [1000, 13, 12] 12)
123 notation (latex output)
124   "return" ("\<^raw:{\textsf{return}}>")
126 translations
127   "_do f" => "f"
128   "_bindM x f g" => "f \<guillemotright>= (\<lambda>x. g)"
129   "_chainM f g" => "f \<guillemotright> g"
130   "_let x t f" => "CONST Let t (\<lambda>x. f)"
131   "_nil f" => "f"
133 print_translation {*
134 let
135   fun dest_abs_eta (Abs (abs as (_, ty, _))) =
136         let
137           val (v, t) = Syntax.variant_abs abs;
138         in (Free (v, ty), t) end
139     | dest_abs_eta t =
140         let
141           val (v, t) = Syntax.variant_abs ("", dummyT, t \$ Bound 0);
142         in (Free (v, dummyT), t) end;
143   fun unfold_monad (Const (@{const_syntax bindM}, _) \$ f \$ g) =
144         let
145           val (v, g') = dest_abs_eta g;
146           val vs = fold_aterms (fn Free (v, _) => insert (op =) v | _ => I) v [];
147           val v_used = fold_aterms
148             (fn Free (w, _) => (fn s => s orelse member (op =) vs w) | _ => I) g' false;
149         in if v_used then
150           Const ("_bindM", dummyT) \$ v \$ f \$ unfold_monad g'
151         else
152           Const ("_chainM", dummyT) \$ f \$ unfold_monad g'
153         end
154     | unfold_monad (Const (@{const_syntax chainM}, _) \$ f \$ g) =
155         Const ("_chainM", dummyT) \$ f \$ unfold_monad g
156     | unfold_monad (Const (@{const_syntax Let}, _) \$ f \$ g) =
157         let
158           val (v, g') = dest_abs_eta g;
159         in Const ("_let", dummyT) \$ v \$ f \$ unfold_monad g' end
160     | unfold_monad (Const (@{const_syntax Pair}, _) \$ f) =
161         Const (@{const_syntax return}, dummyT) \$ f
162     | unfold_monad f = f;
163   fun contains_bindM (Const (@{const_syntax bindM}, _) \$ _ \$ _) = true
164     | contains_bindM (Const (@{const_syntax Let}, _) \$ _ \$ Abs (_, _, t)) =
165         contains_bindM t;
166   fun bindM_monad_tr' (f::g::ts) = list_comb
167     (Const ("_do", dummyT) \$ unfold_monad (Const (@{const_syntax bindM}, dummyT) \$ f \$ g), ts);
168   fun Let_monad_tr' (f :: (g as Abs (_, _, g')) :: ts) = if contains_bindM g' then list_comb
169       (Const ("_do", dummyT) \$ unfold_monad (Const (@{const_syntax Let}, dummyT) \$ f \$ g), ts)
170     else raise Match;
171 in [
174 ] end;
175 *}
178 subsection {* Monad properties *}
180 subsubsection {* Monad laws *}
182 lemma return_bind: "return x \<guillemotright>= f = f x"
183   by (simp add: bindM_def return_def)
185 lemma bind_return: "f \<guillemotright>= return = f"
186 proof (rule Heap_eqI)
187   fix h
188   show "execute (f \<guillemotright>= return) h = execute f h"
189     by (auto simp add: bindM_def return_def split: sum.splits prod.splits)
190 qed
192 lemma bind_bind: "(f \<guillemotright>= g) \<guillemotright>= h = f \<guillemotright>= (\<lambda>x. g x \<guillemotright>= h)"
193   by (rule Heap_eqI) (auto simp add: bindM_def split: split: sum.splits prod.splits)
195 lemma bind_bind': "f \<guillemotright>= (\<lambda>x. g x \<guillemotright>= h x) = f \<guillemotright>= (\<lambda>x. g x \<guillemotright>= (\<lambda>y. return (x, y))) \<guillemotright>= (\<lambda>(x, y). h x y)"
196   by (rule Heap_eqI) (auto simp add: bindM_def split: split: sum.splits prod.splits)
198 lemma raise_bind: "raise e \<guillemotright>= f = raise e"
199   by (simp add: raise_def bindM_def)
202 lemmas monad_simp = return_bind bind_return bind_bind raise_bind
205 subsection {* Generic combinators *}
207 definition
208   liftM :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b Heap"
209 where
210   "liftM f = return o f"
212 definition
213   compM :: "('a \<Rightarrow> 'b Heap) \<Rightarrow> ('b \<Rightarrow> 'c Heap) \<Rightarrow> 'a \<Rightarrow> 'c Heap" (infixl ">>==" 54)
214 where
215   "(f >>== g) = (\<lambda>x. f x \<guillemotright>= g)"
217 notation
218   compM (infixl "\<guillemotright>==" 54)
220 lemma liftM_collapse: "liftM f x = return (f x)"
223 lemma liftM_compM: "liftM f \<guillemotright>== g = g o f"
224   by (auto intro: Heap_eqI' simp add: expand_fun_eq liftM_def compM_def bindM_def)
226 lemma compM_return: "f \<guillemotright>== return = f"
229 lemma compM_compM: "(f \<guillemotright>== g) \<guillemotright>== h = f \<guillemotright>== (g \<guillemotright>== h)"
232 lemma liftM_bind:
233   "(\<lambda>x. liftM f x \<guillemotright>= liftM g) = liftM (\<lambda>x. g (f x))"
236 lemma liftM_comp:
237   "liftM f o g = liftM (f o g)"
238   by (rule Heap_eqI') (simp add: liftM_def)
241   compM_compM liftM_bind liftM_comp
243 primrec
244   mapM :: "('a \<Rightarrow> 'b Heap) \<Rightarrow> 'a list \<Rightarrow> 'b list Heap"
245 where
246   "mapM f [] = return []"
247   | "mapM f (x#xs) = do y \<leftarrow> f x;
248                         ys \<leftarrow> mapM f xs;
249                         return (y # ys)
250                      done"
252 primrec
253   foldM :: "('a \<Rightarrow> 'b \<Rightarrow> 'b Heap) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b Heap"
254 where
255   "foldM f [] s = return s"
256   | "foldM f (x#xs) s = f x s \<guillemotright>= foldM f xs"
258 definition
259   assert :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a Heap"
260 where
261   "assert P x = (if P x then return x else raise (''assert''))"
263 lemma assert_cong [fundef_cong]:
264   assumes "P = P'"
265   assumes "\<And>x. P' x \<Longrightarrow> f x = f' x"
266   shows "(assert P x >>= f) = (assert P' x >>= f')"
267   using assms by (auto simp add: assert_def return_bind raise_bind)
269 subsubsection {* A monadic combinator for simple recursive functions *}
271 function (default "\<lambda>(f,g,x,h). (Inr Exn, undefined)")
272   mrec
273 where
274   "mrec f g x h =
275    (case Heap_Monad.execute (f x) h of
276      (Inl (Inl r), h') \<Rightarrow> (Inl r, h')
277    | (Inl (Inr s), h') \<Rightarrow>
278           (case mrec f g s h' of
279              (Inl z, h'') \<Rightarrow> Heap_Monad.execute (g x s z) h''
280            | (Inr e, h'') \<Rightarrow> (Inr e, h''))
281    | (Inr e, h') \<Rightarrow> (Inr e, h')
282    )"
283 by auto
285 lemma graph_implies_dom:
286 	"mrec_graph x y \<Longrightarrow> mrec_dom x"
287 apply (induct rule:mrec_graph.induct)
288 apply (rule accpI)
289 apply (erule mrec_rel.cases)
290 by simp
292 lemma f_default: "\<not> mrec_dom (f, g, x, h) \<Longrightarrow> mrec f g x h = (Inr Exn, undefined)"
293 	unfolding mrec_def
294   by (rule fundef_default_value[OF mrec_sumC_def graph_implies_dom, of _ _ "(f, g, x, h)", simplified])
296 lemma f_di_reverse:
297   assumes "\<not> mrec_dom (f, g, x, h)"
298   shows "
299    (case Heap_Monad.execute (f x) h of
300      (Inl (Inl r), h') \<Rightarrow> mrecalse
301    | (Inl (Inr s), h') \<Rightarrow> \<not> mrec_dom (f, g, s, h')
302    | (Inr e, h') \<Rightarrow> mrecalse
303    )"
304 using assms
305 by (auto split:prod.splits sum.splits)
306  (erule notE, rule accpI, elim mrec_rel.cases, simp)+
309 lemma mrec_rule:
310   "mrec f g x h =
311    (case Heap_Monad.execute (f x) h of
312      (Inl (Inl r), h') \<Rightarrow> (Inl r, h')
313    | (Inl (Inr s), h') \<Rightarrow>
314           (case mrec f g s h' of
315              (Inl z, h'') \<Rightarrow> Heap_Monad.execute (g x s z) h''
316            | (Inr e, h'') \<Rightarrow> (Inr e, h''))
317    | (Inr e, h') \<Rightarrow> (Inr e, h')
318    )"
319 apply (cases "mrec_dom (f,g,x,h)", simp)
320 apply (frule f_default)
321 apply (frule f_di_reverse, simp)
322 by (auto split: sum.split prod.split simp: f_default)
325 definition
326   "MREC f g x = Heap (mrec f g x)"
328 lemma MREC_rule:
329   "MREC f g x =
330   (do y \<leftarrow> f x;
331                 (case y of
332                 Inl r \<Rightarrow> return r
333               | Inr s \<Rightarrow>
334                 do z \<leftarrow> MREC f g s ;
335                    g x s z
336                 done) done)"
337   unfolding MREC_def
338   unfolding bindM_def return_def
339   apply simp
340   apply (rule ext)
341   apply (unfold mrec_rule[of f g x])
342   by (auto split:prod.splits sum.splits)
344 hide (open) const heap execute
347 subsection {* Code generator setup *}
349 subsubsection {* Logical intermediate layer *}
351 definition
352   Fail :: "String.literal \<Rightarrow> exception"
353 where
354   [code del]: "Fail s = Exn"
356 definition
357   raise_exc :: "exception \<Rightarrow> 'a Heap"
358 where
359   [code del]: "raise_exc e = raise []"
361 lemma raise_raise_exc [code, code_unfold]:
362   "raise s = raise_exc (Fail (STR s))"
363   unfolding Fail_def raise_exc_def raise_def ..
365 hide (open) const Fail raise_exc
368 subsubsection {* SML and OCaml *}
370 code_type Heap (SML "unit/ ->/ _")
371 code_const Heap (SML "raise/ (Fail/ \"bare Heap\")")
372 code_const "op \<guillemotright>=" (SML "!(fn/ f'_/ =>/ fn/ ()/ =>/ f'_/ (_/ ())/ ())")
373 code_const return (SML "!(fn/ ()/ =>/ _)")
375 code_const "Heap_Monad.raise_exc" (SML "!(fn/ ()/ =>/ raise/ _)")
377 code_type Heap (OCaml "_")
378 code_const Heap (OCaml "failwith/ \"bare Heap\"")
379 code_const "op \<guillemotright>=" (OCaml "!(fun/ f'_/ ()/ ->/ f'_/ (_/ ())/ ())")
380 code_const return (OCaml "!(fun/ ()/ ->/ _)")
382 code_const "Heap_Monad.raise_exc" (OCaml "!(fun/ ()/ ->/ raise/ _)")
384 setup {*
386 let
388 open Code_Thingol;
390 fun imp_program naming =
392   let
393     fun is_const c = case lookup_const naming c
394      of SOME c' => (fn c'' => c' = c'')
395       | NONE => K false;
396     val is_bindM = is_const @{const_name bindM};
397     val is_return = is_const @{const_name return};
398     val dummy_name = "";
399     val dummy_type = ITyVar dummy_name;
400     val dummy_case_term = IVar NONE;
401     (*assumption: dummy values are not relevant for serialization*)
402     val unitt = case lookup_const naming @{const_name Unity}
403      of SOME unit' => IConst (unit', (([], []), []))
404       | NONE => error ("Must include " ^ @{const_name Unity} ^ " in generated constants.");
405     fun dest_abs ((v, ty) `|=> t, _) = ((v, ty), t)
406       | dest_abs (t, ty) =
407           let
408             val vs = fold_varnames cons t [];
409             val v = Name.variant vs "x";
410             val ty' = (hd o fst o unfold_fun) ty;
411           in ((SOME v, ty'), t `\$ IVar (SOME v)) end;
412     fun force (t as IConst (c, _) `\$ t') = if is_return c
413           then t' else t `\$ unitt
414       | force t = t `\$ unitt;
415     fun tr_bind' [(t1, _), (t2, ty2)] =
416       let
417         val ((v, ty), t) = dest_abs (t2, ty2);
418       in ICase (((force t1, ty), [(IVar v, tr_bind'' t)]), dummy_case_term) end
419     and tr_bind'' t = case unfold_app t
420          of (IConst (c, (_, ty1 :: ty2 :: _)), [x1, x2]) => if is_bindM c
421               then tr_bind' [(x1, ty1), (x2, ty2)]
422               else force t
423           | _ => force t;
424     fun imp_monad_bind'' ts = (SOME dummy_name, dummy_type) `|=> ICase (((IVar (SOME dummy_name), dummy_type),
425       [(unitt, tr_bind' ts)]), dummy_case_term)
426     and imp_monad_bind' (const as (c, (_, tys))) ts = if is_bindM c then case (ts, tys)
427        of ([t1, t2], ty1 :: ty2 :: _) => imp_monad_bind'' [(t1, ty1), (t2, ty2)]
428         | ([t1, t2, t3], ty1 :: ty2 :: _) => imp_monad_bind'' [(t1, ty1), (t2, ty2)] `\$ t3
429         | (ts, _) => imp_monad_bind (eta_expand 2 (const, ts))
430       else IConst const `\$\$ map imp_monad_bind ts
432       | imp_monad_bind (t as IVar _) = t
433       | imp_monad_bind (t as _ `\$ _) = (case unfold_app t
434          of (IConst const, ts) => imp_monad_bind' const ts
437       | imp_monad_bind (ICase (((t, ty), pats), t0)) = ICase
439             (map o pairself) imp_monad_bind pats),
442   in (Graph.map_nodes o map_terms_stmt) imp_monad_bind end;
444 in
446 Code_Target.extend_target ("SML_imp", ("SML", imp_program))
447 #> Code_Target.extend_target ("OCaml_imp", ("OCaml", imp_program))
449 end
451 *}
453 code_reserved OCaml Failure raise
458 text {* Adaption layer *}
463 import qualified Data.STRef;
464 import qualified Data.Array.ST;
467 type ST s a = Control.Monad.ST.ST s a;
468 type STRef s a = Data.STRef.STRef s a;
469 type STArray s a = Data.Array.ST.STArray s Int a;
471 newSTRef = Data.STRef.newSTRef;
473 writeSTRef = Data.STRef.writeSTRef;
475 newArray :: (Int, Int) -> a -> ST s (STArray s a);
476 newArray = Data.Array.ST.newArray;
478 newListArray :: (Int, Int) -> [a] -> ST s (STArray s a);
479 newListArray = Data.Array.ST.newListArray;
481 lengthArray :: STArray s a -> ST s Int;
482 lengthArray a = Control.Monad.liftM snd (Data.Array.ST.getBounds a);
484 readArray :: STArray s a -> Int -> ST s a;