author wenzelm Thu Oct 16 22:44:24 2008 +0200 (2008-10-16) changeset 28615 4c8fa015ec7f parent 28562 4e74209f113e child 28663 bd8438543bf2 permissions -rw-r--r--
explicit SORT_CONSTRAINT for proofs depending implicitly on certain sorts;
```     1 (*  Title:      HOL/Library/Heap_Monad.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     John Matthews, Galois Connections; Alexander Krauss, Lukas Bulwahn & Florian Haftmann, TU Muenchen
```
```     4 *)
```
```     5
```
```     6 header {* A monad with a polymorphic heap *}
```
```     7
```
```     8 theory Heap_Monad
```
```     9 imports Heap
```
```    10 begin
```
```    11
```
```    12 subsection {* The monad *}
```
```    13
```
```    14 subsubsection {* Monad combinators *}
```
```    15
```
```    16 datatype exception = Exn
```
```    17
```
```    18 text {* Monadic heap actions either produce values
```
```    19   and transform the heap, or fail *}
```
```    20 datatype 'a Heap = Heap "heap \<Rightarrow> ('a + exception) \<times> heap"
```
```    21
```
```    22 primrec
```
```    23   execute :: "'a Heap \<Rightarrow> heap \<Rightarrow> ('a + exception) \<times> heap" where
```
```    24   "execute (Heap f) = f"
```
```    25 lemmas [code del] = execute.simps
```
```    26
```
```    27 lemma Heap_execute [simp]:
```
```    28   "Heap (execute f) = f" by (cases f) simp_all
```
```    29
```
```    30 lemma Heap_eqI:
```
```    31   "(\<And>h. execute f h = execute g h) \<Longrightarrow> f = g"
```
```    32     by (cases f, cases g) (auto simp: expand_fun_eq)
```
```    33
```
```    34 lemma Heap_eqI':
```
```    35   "(\<And>h. (\<lambda>x. execute (f x) h) = (\<lambda>y. execute (g y) h)) \<Longrightarrow> f = g"
```
```    36     by (auto simp: expand_fun_eq intro: Heap_eqI)
```
```    37
```
```    38 lemma Heap_strip: "(\<And>f. PROP P f) \<equiv> (\<And>g. PROP P (Heap g))"
```
```    39 proof
```
```    40   fix g :: "heap \<Rightarrow> ('a + exception) \<times> heap"
```
```    41   assume "\<And>f. PROP P f"
```
```    42   then show "PROP P (Heap g)" .
```
```    43 next
```
```    44   fix f :: "'a Heap"
```
```    45   assume assm: "\<And>g. PROP P (Heap g)"
```
```    46   then have "PROP P (Heap (execute f))" .
```
```    47   then show "PROP P f" by simp
```
```    48 qed
```
```    49
```
```    50 definition
```
```    51   heap :: "(heap \<Rightarrow> 'a \<times> heap) \<Rightarrow> 'a Heap" where
```
```    52   [code del]: "heap f = Heap (\<lambda>h. apfst Inl (f h))"
```
```    53
```
```    54 lemma execute_heap [simp]:
```
```    55   "execute (heap f) h = apfst Inl (f h)"
```
```    56   by (simp add: heap_def)
```
```    57
```
```    58 definition
```
```    59   bindM :: "'a Heap \<Rightarrow> ('a \<Rightarrow> 'b Heap) \<Rightarrow> 'b Heap" (infixl ">>=" 54) where
```
```    60   [code del]: "f >>= g = Heap (\<lambda>h. case execute f h of
```
```    61                   (Inl x, h') \<Rightarrow> execute (g x) h'
```
```    62                 | r \<Rightarrow> r)"
```
```    63
```
```    64 notation
```
```    65   bindM (infixl "\<guillemotright>=" 54)
```
```    66
```
```    67 abbreviation
```
```    68   chainM :: "'a Heap \<Rightarrow> 'b Heap \<Rightarrow> 'b Heap"  (infixl ">>" 54) where
```
```    69   "f >> g \<equiv> f >>= (\<lambda>_. g)"
```
```    70
```
```    71 notation
```
```    72   chainM (infixl "\<guillemotright>" 54)
```
```    73
```
```    74 definition
```
```    75   return :: "'a \<Rightarrow> 'a Heap" where
```
```    76   [code del]: "return x = heap (Pair x)"
```
```    77
```
```    78 lemma execute_return [simp]:
```
```    79   "execute (return x) h = apfst Inl (x, h)"
```
```    80   by (simp add: return_def)
```
```    81
```
```    82 definition
```
```    83   raise :: "string \<Rightarrow> 'a Heap" where -- {* the string is just decoration *}
```
```    84   [code del]: "raise s = Heap (Pair (Inr Exn))"
```
```    85
```
```    86 notation (latex output)
```
```    87   "raise" ("\<^raw:{\textsf{raise}}>")
```
```    88
```
```    89 lemma execute_raise [simp]:
```
```    90   "execute (raise s) h = (Inr Exn, h)"
```
```    91   by (simp add: raise_def)
```
```    92
```
```    93
```
```    94 subsubsection {* do-syntax *}
```
```    95
```
```    96 text {*
```
```    97   We provide a convenient do-notation for monadic expressions
```
```    98   well-known from Haskell.  @{const Let} is printed
```
```    99   specially in do-expressions.
```
```   100 *}
```
```   101
```
```   102 nonterminals do_expr
```
```   103
```
```   104 syntax
```
```   105   "_do" :: "do_expr \<Rightarrow> 'a"
```
```   106     ("(do (_)//done)" [12] 100)
```
```   107   "_bindM" :: "pttrn \<Rightarrow> 'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
```
```   108     ("_ <- _;//_" [1000, 13, 12] 12)
```
```   109   "_chainM" :: "'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
```
```   110     ("_;//_" [13, 12] 12)
```
```   111   "_let" :: "pttrn \<Rightarrow> 'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
```
```   112     ("let _ = _;//_" [1000, 13, 12] 12)
```
```   113   "_nil" :: "'a \<Rightarrow> do_expr"
```
```   114     ("_" [12] 12)
```
```   115
```
```   116 syntax (xsymbols)
```
```   117   "_bindM" :: "pttrn \<Rightarrow> 'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
```
```   118     ("_ \<leftarrow> _;//_" [1000, 13, 12] 12)
```
```   119 syntax (latex output)
```
```   120   "_do" :: "do_expr \<Rightarrow> 'a"
```
```   121     ("(\<^raw:{\textsf{do}}> (_))" [12] 100)
```
```   122   "_let" :: "pttrn \<Rightarrow> 'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
```
```   123     ("\<^raw:\textsf{let}> _ = _;//_" [1000, 13, 12] 12)
```
```   124 notation (latex output)
```
```   125   "return" ("\<^raw:{\textsf{return}}>")
```
```   126
```
```   127 translations
```
```   128   "_do f" => "f"
```
```   129   "_bindM x f g" => "f \<guillemotright>= (\<lambda>x. g)"
```
```   130   "_chainM f g" => "f \<guillemotright> g"
```
```   131   "_let x t f" => "CONST Let t (\<lambda>x. f)"
```
```   132   "_nil f" => "f"
```
```   133
```
```   134 print_translation {*
```
```   135 let
```
```   136   fun dest_abs_eta (Abs (abs as (_, ty, _))) =
```
```   137         let
```
```   138           val (v, t) = Syntax.variant_abs abs;
```
```   139         in (Free (v, ty), t) end
```
```   140     | dest_abs_eta t =
```
```   141         let
```
```   142           val (v, t) = Syntax.variant_abs ("", dummyT, t \$ Bound 0);
```
```   143         in (Free (v, dummyT), t) end;
```
```   144   fun unfold_monad (Const (@{const_syntax bindM}, _) \$ f \$ g) =
```
```   145         let
```
```   146           val (v, g') = dest_abs_eta g;
```
```   147           val vs = fold_aterms (fn Free (v, _) => insert (op =) v | _ => I) v [];
```
```   148           val v_used = fold_aterms
```
```   149             (fn Free (w, _) => (fn s => s orelse member (op =) vs w) | _ => I) g' false;
```
```   150         in if v_used then
```
```   151           Const ("_bindM", dummyT) \$ v \$ f \$ unfold_monad g'
```
```   152         else
```
```   153           Const ("_chainM", dummyT) \$ f \$ unfold_monad g'
```
```   154         end
```
```   155     | unfold_monad (Const (@{const_syntax chainM}, _) \$ f \$ g) =
```
```   156         Const ("_chainM", dummyT) \$ f \$ unfold_monad g
```
```   157     | unfold_monad (Const (@{const_syntax Let}, _) \$ f \$ g) =
```
```   158         let
```
```   159           val (v, g') = dest_abs_eta g;
```
```   160         in Const ("_let", dummyT) \$ v \$ f \$ unfold_monad g' end
```
```   161     | unfold_monad (Const (@{const_syntax Pair}, _) \$ f) =
```
```   162         Const (@{const_syntax return}, dummyT) \$ f
```
```   163     | unfold_monad f = f;
```
```   164   fun contains_bindM (Const (@{const_syntax bindM}, _) \$ _ \$ _) = true
```
```   165     | contains_bindM (Const (@{const_syntax Let}, _) \$ _ \$ Abs (_, _, t)) =
```
```   166         contains_bindM t;
```
```   167   fun bindM_monad_tr' (f::g::ts) = list_comb
```
```   168     (Const ("_do", dummyT) \$ unfold_monad (Const (@{const_syntax bindM}, dummyT) \$ f \$ g), ts);
```
```   169   fun Let_monad_tr' (f :: (g as Abs (_, _, g')) :: ts) = if contains_bindM g' then list_comb
```
```   170       (Const ("_do", dummyT) \$ unfold_monad (Const (@{const_syntax Let}, dummyT) \$ f \$ g), ts)
```
```   171     else raise Match;
```
```   172 in [
```
```   173   (@{const_syntax bindM}, bindM_monad_tr'),
```
```   174   (@{const_syntax Let}, Let_monad_tr')
```
```   175 ] end;
```
```   176 *}
```
```   177
```
```   178
```
```   179 subsection {* Monad properties *}
```
```   180
```
```   181 subsubsection {* Monad laws *}
```
```   182
```
```   183 lemma return_bind: "return x \<guillemotright>= f = f x"
```
```   184   by (simp add: bindM_def return_def)
```
```   185
```
```   186 lemma bind_return: "f \<guillemotright>= return = f"
```
```   187 proof (rule Heap_eqI)
```
```   188   fix h
```
```   189   show "execute (f \<guillemotright>= return) h = execute f h"
```
```   190     by (auto simp add: bindM_def return_def split: sum.splits prod.splits)
```
```   191 qed
```
```   192
```
```   193 lemma bind_bind: "(f \<guillemotright>= g) \<guillemotright>= h = f \<guillemotright>= (\<lambda>x. g x \<guillemotright>= h)"
```
```   194   by (rule Heap_eqI) (auto simp add: bindM_def split: split: sum.splits prod.splits)
```
```   195
```
```   196 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)"
```
```   197   by (rule Heap_eqI) (auto simp add: bindM_def split: split: sum.splits prod.splits)
```
```   198
```
```   199 lemma raise_bind: "raise e \<guillemotright>= f = raise e"
```
```   200   by (simp add: raise_def bindM_def)
```
```   201
```
```   202
```
```   203 lemmas monad_simp = return_bind bind_return bind_bind raise_bind
```
```   204
```
```   205
```
```   206 subsection {* Generic combinators *}
```
```   207
```
```   208 definition
```
```   209   liftM :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b Heap"
```
```   210 where
```
```   211   "liftM f = return o f"
```
```   212
```
```   213 definition
```
```   214   compM :: "('a \<Rightarrow> 'b Heap) \<Rightarrow> ('b \<Rightarrow> 'c Heap) \<Rightarrow> 'a \<Rightarrow> 'c Heap" (infixl ">>==" 54)
```
```   215 where
```
```   216   "(f >>== g) = (\<lambda>x. f x \<guillemotright>= g)"
```
```   217
```
```   218 notation
```
```   219   compM (infixl "\<guillemotright>==" 54)
```
```   220
```
```   221 lemma liftM_collapse: "liftM f x = return (f x)"
```
```   222   by (simp add: liftM_def)
```
```   223
```
```   224 lemma liftM_compM: "liftM f \<guillemotright>== g = g o f"
```
```   225   by (auto intro: Heap_eqI' simp add: expand_fun_eq liftM_def compM_def bindM_def)
```
```   226
```
```   227 lemma compM_return: "f \<guillemotright>== return = f"
```
```   228   by (simp add: compM_def monad_simp)
```
```   229
```
```   230 lemma compM_compM: "(f \<guillemotright>== g) \<guillemotright>== h = f \<guillemotright>== (g \<guillemotright>== h)"
```
```   231   by (simp add: compM_def monad_simp)
```
```   232
```
```   233 lemma liftM_bind:
```
```   234   "(\<lambda>x. liftM f x \<guillemotright>= liftM g) = liftM (\<lambda>x. g (f x))"
```
```   235   by (rule Heap_eqI') (simp add: monad_simp liftM_def bindM_def)
```
```   236
```
```   237 lemma liftM_comp:
```
```   238   "liftM f o g = liftM (f o g)"
```
```   239   by (rule Heap_eqI') (simp add: liftM_def)
```
```   240
```
```   241 lemmas monad_simp' = monad_simp liftM_compM compM_return
```
```   242   compM_compM liftM_bind liftM_comp
```
```   243
```
```   244 primrec
```
```   245   mapM :: "('a \<Rightarrow> 'b Heap) \<Rightarrow> 'a list \<Rightarrow> 'b list Heap"
```
```   246 where
```
```   247   "mapM f [] = return []"
```
```   248   | "mapM f (x#xs) = do y \<leftarrow> f x;
```
```   249                         ys \<leftarrow> mapM f xs;
```
```   250                         return (y # ys)
```
```   251                      done"
```
```   252
```
```   253 primrec
```
```   254   foldM :: "('a \<Rightarrow> 'b \<Rightarrow> 'b Heap) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b Heap"
```
```   255 where
```
```   256   "foldM f [] s = return s"
```
```   257   | "foldM f (x#xs) s = f x s \<guillemotright>= foldM f xs"
```
```   258
```
```   259 hide (open) const heap execute
```
```   260
```
```   261
```
```   262 subsection {* Code generator setup *}
```
```   263
```
```   264 subsubsection {* Logical intermediate layer *}
```
```   265
```
```   266 definition
```
```   267   Fail :: "message_string \<Rightarrow> exception"
```
```   268 where
```
```   269   [code del]: "Fail s = Exn"
```
```   270
```
```   271 definition
```
```   272   raise_exc :: "exception \<Rightarrow> 'a Heap"
```
```   273 where
```
```   274   [code del]: "raise_exc e = raise []"
```
```   275
```
```   276 lemma raise_raise_exc [code, code inline]:
```
```   277   "raise s = raise_exc (Fail (STR s))"
```
```   278   unfolding Fail_def raise_exc_def raise_def ..
```
```   279
```
```   280 hide (open) const Fail raise_exc
```
```   281
```
```   282
```
```   283 subsubsection {* SML and OCaml *}
```
```   284
```
```   285 code_type Heap (SML "unit/ ->/ _")
```
```   286 code_const Heap (SML "raise/ (Fail/ \"bare Heap\")")
```
```   287 code_const "op \<guillemotright>=" (SML "!(fn/ f'_/ =>/ fn/ ()/ =>/ f'_/ (_/ ())/ ())")
```
```   288 code_const return (SML "!(fn/ ()/ =>/ _)")
```
```   289 code_const "Heap_Monad.Fail" (SML "Fail")
```
```   290 code_const "Heap_Monad.raise_exc" (SML "!(fn/ ()/ =>/ raise/ _)")
```
```   291
```
```   292 code_type Heap (OCaml "_")
```
```   293 code_const Heap (OCaml "failwith/ \"bare Heap\"")
```
```   294 code_const "op \<guillemotright>=" (OCaml "!(fun/ f'_/ ()/ ->/ f'_/ (_/ ())/ ())")
```
```   295 code_const return (OCaml "!(fun/ ()/ ->/ _)")
```
```   296 code_const "Heap_Monad.Fail" (OCaml "Failure")
```
```   297 code_const "Heap_Monad.raise_exc" (OCaml "!(fun/ ()/ ->/ raise/ _)")
```
```   298
```
```   299 ML {*
```
```   300 local
```
```   301
```
```   302 open Code_Thingol;
```
```   303
```
```   304 val bind' = Code_Name.const @{theory} @{const_name bindM};
```
```   305 val return' = Code_Name.const @{theory} @{const_name return};
```
```   306 val unit' = Code_Name.const @{theory} @{const_name Unity};
```
```   307
```
```   308 fun imp_monad_bind'' ts =
```
```   309   let
```
```   310     val dummy_name = "";
```
```   311     val dummy_type = ITyVar dummy_name;
```
```   312     val dummy_case_term = IVar dummy_name;
```
```   313     (*assumption: dummy values are not relevant for serialization*)
```
```   314     val unitt = IConst (unit', ([], []));
```
```   315     fun dest_abs ((v, ty) `|-> t, _) = ((v, ty), t)
```
```   316       | dest_abs (t, ty) =
```
```   317           let
```
```   318             val vs = Code_Thingol.fold_varnames cons t [];
```
```   319             val v = Name.variant vs "x";
```
```   320             val ty' = (hd o fst o Code_Thingol.unfold_fun) ty;
```
```   321           in ((v, ty'), t `\$ IVar v) end;
```
```   322     fun force (t as IConst (c, _) `\$ t') = if c = return'
```
```   323           then t' else t `\$ unitt
```
```   324       | force t = t `\$ unitt;
```
```   325     fun tr_bind' [(t1, _), (t2, ty2)] =
```
```   326       let
```
```   327         val ((v, ty), t) = dest_abs (t2, ty2);
```
```   328       in ICase (((force t1, ty), [(IVar v, tr_bind'' t)]), dummy_case_term) end
```
```   329     and tr_bind'' t = case Code_Thingol.unfold_app t
```
```   330          of (IConst (c, (_, ty1 :: ty2 :: _)), [x1, x2]) => if c = bind'
```
```   331               then tr_bind' [(x1, ty1), (x2, ty2)]
```
```   332               else force t
```
```   333           | _ => force t;
```
```   334   in (dummy_name, dummy_type) `|-> ICase (((IVar dummy_name, dummy_type),
```
```   335     [(unitt, tr_bind' ts)]), dummy_case_term) end
```
```   336 and imp_monad_bind' (const as (c, (_, tys))) ts = if c = bind' then case (ts, tys)
```
```   337    of ([t1, t2], ty1 :: ty2 :: _) => imp_monad_bind'' [(t1, ty1), (t2, ty2)]
```
```   338     | ([t1, t2, t3], ty1 :: ty2 :: _) => imp_monad_bind'' [(t1, ty1), (t2, ty2)] `\$ t3
```
```   339     | (ts, _) => imp_monad_bind (eta_expand 2 (const, ts))
```
```   340   else IConst const `\$\$ map imp_monad_bind ts
```
```   341 and imp_monad_bind (IConst const) = imp_monad_bind' const []
```
```   342   | imp_monad_bind (t as IVar _) = t
```
```   343   | imp_monad_bind (t as _ `\$ _) = (case unfold_app t
```
```   344      of (IConst const, ts) => imp_monad_bind' const ts
```
```   345       | (t, ts) => imp_monad_bind t `\$\$ map imp_monad_bind ts)
```
```   346   | imp_monad_bind (v_ty `|-> t) = v_ty `|-> imp_monad_bind t
```
```   347   | imp_monad_bind (ICase (((t, ty), pats), t0)) = ICase
```
```   348       (((imp_monad_bind t, ty), (map o pairself) imp_monad_bind pats), imp_monad_bind t0);
```
```   349
```
```   350 in
```
```   351
```
```   352 val imp_program = (Graph.map_nodes o map_terms_stmt) imp_monad_bind;
```
```   353
```
```   354 end
```
```   355 *}
```
```   356
```
```   357 setup {* Code_Target.extend_target ("SML_imp", ("SML", imp_program)) *}
```
```   358 setup {* Code_Target.extend_target ("OCaml_imp", ("OCaml", imp_program)) *}
```
```   359
```
```   360 code_reserved OCaml Failure raise
```
```   361
```
```   362
```
```   363 subsubsection {* Haskell *}
```
```   364
```
```   365 text {* Adaption layer *}
```
```   366
```
```   367 code_include Haskell "STMonad"
```
```   368 {*import qualified Control.Monad;
```
```   369 import qualified Control.Monad.ST;
```
```   370 import qualified Data.STRef;
```
```   371 import qualified Data.Array.ST;
```
```   372
```
```   373 type RealWorld = Control.Monad.ST.RealWorld;
```
```   374 type ST s a = Control.Monad.ST.ST s a;
```
```   375 type STRef s a = Data.STRef.STRef s a;
```
```   376 type STArray s a = Data.Array.ST.STArray s Int a;
```
```   377
```
```   378 runST :: (forall s. ST s a) -> a;
```
```   379 runST s = Control.Monad.ST.runST s;
```
```   380
```
```   381 newSTRef = Data.STRef.newSTRef;
```
```   382 readSTRef = Data.STRef.readSTRef;
```
```   383 writeSTRef = Data.STRef.writeSTRef;
```
```   384
```
```   385 newArray :: (Int, Int) -> a -> ST s (STArray s a);
```
```   386 newArray = Data.Array.ST.newArray;
```
```   387
```
```   388 newListArray :: (Int, Int) -> [a] -> ST s (STArray s a);
```
```   389 newListArray = Data.Array.ST.newListArray;
```
```   390
```
```   391 lengthArray :: STArray s a -> ST s Int;
```
```   392 lengthArray a = Control.Monad.liftM snd (Data.Array.ST.getBounds a);
```
```   393
```
```   394 readArray :: STArray s a -> Int -> ST s a;
```
```   395 readArray = Data.Array.ST.readArray;
```
```   396
```
```   397 writeArray :: STArray s a -> Int -> a -> ST s ();
```
```   398 writeArray = Data.Array.ST.writeArray;*}
```
```   399
```
```   400 code_reserved Haskell RealWorld ST STRef Array
```
```   401   runST
```
```   402   newSTRef reasSTRef writeSTRef
```
```   403   newArray newListArray lengthArray readArray writeArray
```
```   404
```
```   405 text {* Monad *}
```
```   406
```
```   407 code_type Heap (Haskell "ST/ RealWorld/ _")
```
```   408 code_const Heap (Haskell "error/ \"bare Heap\"")
```
```   409 code_monad "op \<guillemotright>=" Haskell
```
```   410 code_const return (Haskell "return")
```
```   411 code_const "Heap_Monad.Fail" (Haskell "_")
```
```   412 code_const "Heap_Monad.raise_exc" (Haskell "error")
```
```   413
```
```   414 end
```