
haftmann@26170: (*  Title:      HOL/Library/Heap_Monad.thy
haftmann@26170:     ID:         $Id$
haftmann@26170:     Author:     John Matthews, Galois Connections; Alexander Krauss, Lukas Bulwahn & Florian Haftmann, TU Muenchen
haftmann@26170: *)
haftmann@26170: 
haftmann@26170: header {* A monad with a polymorphic heap *}
haftmann@26170: 
haftmann@26170: theory Heap_Monad
haftmann@26170: imports Heap
haftmann@26170: begin
haftmann@26170: 
haftmann@26170: subsection {* The monad *}
haftmann@26170: 
haftmann@26170: subsubsection {* Monad combinators *}
haftmann@26170: 
haftmann@26170: datatype exception = Exn
haftmann@26170: 
haftmann@26170: text {* Monadic heap actions either produce values
haftmann@26170:   and transform the heap, or fail *}
haftmann@26170: datatype 'a Heap = Heap "heap \<Rightarrow> ('a + exception) \<times> heap"
haftmann@26170: 
haftmann@26170: primrec
haftmann@26170:   execute :: "'a Heap \<Rightarrow> heap \<Rightarrow> ('a + exception) \<times> heap" where
haftmann@26170:   "execute (Heap f) = f"
haftmann@26170: lemmas [code del] = execute.simps
haftmann@26170: 
haftmann@26170: lemma Heap_execute [simp]:
haftmann@26170:   "Heap (execute f) = f" by (cases f) simp_all
haftmann@26170: 
haftmann@26170: lemma Heap_eqI:
haftmann@26170:   "(\<And>h. execute f h = execute g h) \<Longrightarrow> f = g"
haftmann@26170:     by (cases f, cases g) (auto simp: expand_fun_eq)
haftmann@26170: 
haftmann@26170: lemma Heap_eqI':
haftmann@26170:   "(\<And>h. (\<lambda>x. execute (f x) h) = (\<lambda>y. execute (g y) h)) \<Longrightarrow> f = g"
haftmann@26170:     by (auto simp: expand_fun_eq intro: Heap_eqI)
haftmann@26170: 
haftmann@26170: lemma Heap_strip: "(\<And>f. PROP P f) \<equiv> (\<And>g. PROP P (Heap g))"
haftmann@26170: proof
haftmann@26170:   fix g :: "heap \<Rightarrow> ('a + exception) \<times> heap" 
haftmann@26170:   assume "\<And>f. PROP P f"
haftmann@26170:   then show "PROP P (Heap g)" .
haftmann@26170: next
haftmann@26170:   fix f :: "'a Heap" 
haftmann@26170:   assume assm: "\<And>g. PROP P (Heap g)"
haftmann@26170:   then have "PROP P (Heap (execute f))" .
haftmann@26170:   then show "PROP P f" by simp
haftmann@26170: qed
haftmann@26170: 
haftmann@26170: definition
haftmann@26170:   heap :: "(heap \<Rightarrow> 'a \<times> heap) \<Rightarrow> 'a Heap" where
haftmann@26170:   [code del]: "heap f = Heap (\<lambda>h. apfst Inl (f h))"
haftmann@26170: 
haftmann@26170: lemma execute_heap [simp]:
haftmann@26170:   "execute (heap f) h = apfst Inl (f h)"
haftmann@26170:   by (simp add: heap_def)
haftmann@26170: 
haftmann@26170: definition
haftmann@26170:   run :: "'a Heap \<Rightarrow> 'a Heap" where
haftmann@26170:   run_drop [code del]: "run f = f"
haftmann@26170: 
haftmann@26170: definition
haftmann@26170:   bindM :: "'a Heap \<Rightarrow> ('a \<Rightarrow> 'b Heap) \<Rightarrow> 'b Heap" (infixl ">>=" 54) where
haftmann@26170:   [code del]: "f >>= g = Heap (\<lambda>h. case execute f h of
haftmann@26170:                   (Inl x, h') \<Rightarrow> execute (g x) h'
haftmann@26170:                 | r \<Rightarrow> r)"
haftmann@26170: 
haftmann@26170: notation
haftmann@26170:   bindM (infixl "\<guillemotright>=" 54)
haftmann@26170: 
haftmann@26170: abbreviation
haftmann@26170:   chainM :: "'a Heap \<Rightarrow> 'b Heap \<Rightarrow> 'b Heap"  (infixl ">>" 54) where
haftmann@26170:   "f >> g \<equiv> f >>= (\<lambda>_. g)"
haftmann@26170: 
haftmann@26170: notation
haftmann@26170:   chainM (infixl "\<guillemotright>" 54)
haftmann@26170: 
haftmann@26170: definition
haftmann@26170:   return :: "'a \<Rightarrow> 'a Heap" where
haftmann@26170:   [code del]: "return x = heap (Pair x)"
haftmann@26170: 
haftmann@26170: lemma execute_return [simp]:
haftmann@26170:   "execute (return x) h = apfst Inl (x, h)"
haftmann@26170:   by (simp add: return_def)
haftmann@26170: 
haftmann@26170: definition
haftmann@26170:   raise :: "string \<Rightarrow> 'a Heap" where -- {* the string is just decoration *}
haftmann@26170:   [code del]: "raise s = Heap (Pair (Inr Exn))"
haftmann@26170: 
haftmann@26170: notation (latex output)
haftmann@26170:   "raise" ("\<^raw:{\textsf{raise}}>")
haftmann@26170: 
haftmann@26170: lemma execute_raise [simp]:
haftmann@26170:   "execute (raise s) h = (Inr Exn, h)"
haftmann@26170:   by (simp add: raise_def)
haftmann@26170: 
haftmann@26170: 
haftmann@26170: subsubsection {* do-syntax *}
haftmann@26170: 
haftmann@26170: text {*
haftmann@26170:   We provide a convenient do-notation for monadic expressions
haftmann@26170:   well-known from Haskell.  @{const Let} is printed
haftmann@26170:   specially in do-expressions.
haftmann@26170: *}
haftmann@26170: 
haftmann@26170: nonterminals do_expr
haftmann@26170: 
haftmann@26170: syntax
haftmann@26170:   "_do" :: "do_expr \<Rightarrow> 'a"
haftmann@26170:     ("(do (_)//done)" [12] 100)
haftmann@26170:   "_bindM" :: "pttrn \<Rightarrow> 'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
haftmann@26170:     ("_ <- _;//_" [1000, 13, 12] 12)
haftmann@26170:   "_chainM" :: "'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
haftmann@26170:     ("_;//_" [13, 12] 12)
haftmann@26170:   "_let" :: "pttrn \<Rightarrow> 'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
haftmann@26170:     ("let _ = _;//_" [1000, 13, 12] 12)
haftmann@26170:   "_nil" :: "'a \<Rightarrow> do_expr"
haftmann@26170:     ("_" [12] 12)
haftmann@26170: 
haftmann@26170: syntax (xsymbols)
haftmann@26170:   "_bindM" :: "pttrn \<Rightarrow> 'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
haftmann@26170:     ("_ \<leftarrow> _;//_" [1000, 13, 12] 12)
haftmann@26170: syntax (latex output)
haftmann@26170:   "_do" :: "do_expr \<Rightarrow> 'a"
haftmann@26170:     ("(\<^raw:{\textsf{do}}> (_))" [12] 100)
haftmann@26170:   "_let" :: "pttrn \<Rightarrow> 'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
haftmann@26170:     ("\<^raw:\textsf{let}> _ = _;//_" [1000, 13, 12] 12)
haftmann@26170: notation (latex output)
haftmann@26170:   "return" ("\<^raw:{\textsf{return}}>")
haftmann@26170: 
haftmann@26170: translations
haftmann@26170:   "_do f" => "CONST run f"
haftmann@26170:   "_bindM x f g" => "f \<guillemotright>= (\<lambda>x. g)"
haftmann@26170:   "_chainM f g" => "f \<guillemotright> g"
haftmann@26170:   "_let x t f" => "CONST Let t (\<lambda>x. f)"
haftmann@26170:   "_nil f" => "f"
haftmann@26170: 
haftmann@26170: print_translation {*
haftmann@26170: let
haftmann@26170:   fun dest_abs_eta (Abs (abs as (_, ty, _))) =
haftmann@26170:         let
haftmann@26170:           val (v, t) = Syntax.variant_abs abs;
haftmann@26170:         in ((v, ty), t) end
haftmann@26170:     | dest_abs_eta t =
haftmann@26170:         let
haftmann@26170:           val (v, t) = Syntax.variant_abs ("", dummyT, t $ Bound 0);
haftmann@26170:         in ((v, dummyT), t) end
haftmann@26170:   fun unfold_monad (Const (@{const_syntax bindM}, _) $ f $ g) =
haftmann@26170:         let
haftmann@26170:           val ((v, ty), g') = dest_abs_eta g;
haftmann@26170:           val v_used = fold_aterms
haftmann@26170:             (fn Free (w, _) => (fn s => s orelse v = w) | _ => I) g' false;
haftmann@26170:         in if v_used then
haftmann@26170:           Const ("_bindM", dummyT) $ Free (v, ty) $ f $ unfold_monad g'
haftmann@26170:         else
haftmann@26170:           Const ("_chainM", dummyT) $ f $ unfold_monad g'
haftmann@26170:         end
haftmann@26170:     | unfold_monad (Const (@{const_syntax chainM}, _) $ f $ g) =
haftmann@26170:         Const ("_chainM", dummyT) $ f $ unfold_monad g
haftmann@26170:     | unfold_monad (Const (@{const_syntax Let}, _) $ f $ g) =
haftmann@26170:         let
haftmann@26170:           val ((v, ty), g') = dest_abs_eta g;
haftmann@26170:         in Const ("_let", dummyT) $ Free (v, ty) $ f $ unfold_monad g' end
haftmann@26170:     | unfold_monad (Const (@{const_syntax Pair}, _) $ f) =
haftmann@26170:         Const ("return", dummyT) $ f
haftmann@26170:     | unfold_monad f = f;
haftmann@26170:   fun tr' (f::ts) =
haftmann@26170:     list_comb (Const ("_do", dummyT) $ unfold_monad f, ts)
haftmann@26170: in [(@{const_syntax "run"}, tr')] end;
haftmann@26170: *}
haftmann@26170: 
haftmann@26170: 
haftmann@26170: subsection {* Monad properties *}
haftmann@26170: 
haftmann@26170: subsubsection {* Superfluous runs *}
haftmann@26170: 
haftmann@26170: text {* @{term run} is just a doodle *}
haftmann@26170: 
haftmann@26170: lemma run_simp [simp]:
haftmann@26170:   "\<And>f. run (run f) = run f"
haftmann@26170:   "\<And>f g. run f \<guillemotright>= g = f \<guillemotright>= g"
haftmann@26170:   "\<And>f g. run f \<guillemotright> g = f \<guillemotright> g"
haftmann@26170:   "\<And>f g. f \<guillemotright>= (\<lambda>x. run g) = f \<guillemotright>= (\<lambda>x. g)"
haftmann@26170:   "\<And>f g. f \<guillemotright> run g = f \<guillemotright> g"
haftmann@26170:   "\<And>f. f = run g \<longleftrightarrow> f = g"
haftmann@26170:   "\<And>f. run f = g \<longleftrightarrow> f = g"
haftmann@26170:   unfolding run_drop by rule+
haftmann@26170: 
haftmann@26170: subsubsection {* Monad laws *}
haftmann@26170: 
haftmann@26170: lemma return_bind: "return x \<guillemotright>= f = f x"
haftmann@26170:   by (simp add: bindM_def return_def)
haftmann@26170: 
haftmann@26170: lemma bind_return: "f \<guillemotright>= return = f"
haftmann@26170: proof (rule Heap_eqI)
haftmann@26170:   fix h
haftmann@26170:   show "execute (f \<guillemotright>= return) h = execute f h"
haftmann@26170:     by (auto simp add: bindM_def return_def split: sum.splits prod.splits)
haftmann@26170: qed
haftmann@26170: 
haftmann@26170: lemma bind_bind: "(f \<guillemotright>= g) \<guillemotright>= h = f \<guillemotright>= (\<lambda>x. g x \<guillemotright>= h)"
haftmann@26170:   by (rule Heap_eqI) (auto simp add: bindM_def split: split: sum.splits prod.splits)
haftmann@26170: 
haftmann@26170: 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)"
haftmann@26170:   by (rule Heap_eqI) (auto simp add: bindM_def split: split: sum.splits prod.splits)
haftmann@26170: 
haftmann@26170: lemma raise_bind: "raise e \<guillemotright>= f = raise e"
haftmann@26170:   by (simp add: raise_def bindM_def)
haftmann@26170: 
haftmann@26170: 
haftmann@26170: lemmas monad_simp = return_bind bind_return bind_bind raise_bind
haftmann@26170: 
haftmann@26170: 
haftmann@26170: subsection {* Generic combinators *}
haftmann@26170: 
haftmann@26170: definition
haftmann@26170:   liftM :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b Heap"
haftmann@26170: where
haftmann@26170:   "liftM f = return o f"
haftmann@26170: 
haftmann@26170: definition
haftmann@26170:   compM :: "('a \<Rightarrow> 'b Heap) \<Rightarrow> ('b \<Rightarrow> 'c Heap) \<Rightarrow> 'a \<Rightarrow> 'c Heap" (infixl ">>==" 54)
haftmann@26170: where
haftmann@26170:   "(f >>== g) = (\<lambda>x. f x \<guillemotright>= g)"
haftmann@26170: 
haftmann@26170: notation
haftmann@26170:   compM (infixl "\<guillemotright>==" 54)
haftmann@26170: 
haftmann@26170: lemma liftM_collapse: "liftM f x = return (f x)"
haftmann@26170:   by (simp add: liftM_def)
haftmann@26170: 
haftmann@26170: lemma liftM_compM: "liftM f \<guillemotright>== g = g o f"
haftmann@26170:   by (auto intro: Heap_eqI' simp add: expand_fun_eq liftM_def compM_def bindM_def)
haftmann@26170: 
haftmann@26170: lemma compM_return: "f \<guillemotright>== return = f"
haftmann@26170:   by (simp add: compM_def monad_simp)
haftmann@26170: 
haftmann@26170: lemma compM_compM: "(f \<guillemotright>== g) \<guillemotright>== h = f \<guillemotright>== (g \<guillemotright>== h)"
haftmann@26170:   by (simp add: compM_def monad_simp)
haftmann@26170: 
haftmann@26170: lemma liftM_bind:
haftmann@26170:   "(\<lambda>x. liftM f x \<guillemotright>= liftM g) = liftM (\<lambda>x. g (f x))"
haftmann@26170:   by (rule Heap_eqI') (simp add: monad_simp liftM_def bindM_def)
haftmann@26170: 
haftmann@26170: lemma liftM_comp:
haftmann@26170:   "liftM f o g = liftM (f o g)"
haftmann@26170:   by (rule Heap_eqI') (simp add: liftM_def)
haftmann@26170: 
haftmann@26170: lemmas monad_simp' = monad_simp liftM_compM compM_return
haftmann@26170:   compM_compM liftM_bind liftM_comp
haftmann@26170: 
haftmann@26170: primrec 
haftmann@26170:   mapM :: "('a \<Rightarrow> 'b Heap) \<Rightarrow> 'a list \<Rightarrow> 'b list Heap"
haftmann@26170: where
haftmann@26170:   "mapM f [] = return []"
haftmann@26170:   | "mapM f (x#xs) = do y \<leftarrow> f x;
haftmann@26170:                         ys \<leftarrow> mapM f xs;
haftmann@26170:                         return (y # ys)
haftmann@26170:                      done"
haftmann@26170: 
haftmann@26170: primrec
haftmann@26170:   foldM :: "('a \<Rightarrow> 'b \<Rightarrow> 'b Heap) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b Heap"
haftmann@26170: where
haftmann@26170:   "foldM f [] s = return s"
haftmann@26170:   | "foldM f (x#xs) s = f x s \<guillemotright>= foldM f xs"
haftmann@26170: 
haftmann@26170: hide (open) const heap execute
haftmann@26170: 
haftmann@26182: 
haftmann@26182: subsection {* Code generator setup *}
haftmann@26182: 
haftmann@26182: subsubsection {* Logical intermediate layer *}
haftmann@26182: 
haftmann@26182: definition
haftmann@26182:   Fail :: "message_string \<Rightarrow> exception"
haftmann@26182: where
haftmann@26182:   [code func del]: "Fail s = Exn"
haftmann@26182: 
haftmann@26182: definition
haftmann@26182:   raise_exc :: "exception \<Rightarrow> 'a Heap"
haftmann@26182: where
haftmann@26182:   [code func del]: "raise_exc e = raise []"
haftmann@26182: 
haftmann@26182: lemma raise_raise_exc [code func, code inline]:
haftmann@26182:   "raise s = raise_exc (Fail (STR s))"
haftmann@26182:   unfolding Fail_def raise_exc_def raise_def ..
haftmann@26182: 
haftmann@26182: hide (open) const Fail raise_exc
haftmann@26182: 
haftmann@26182: 
haftmann@26182: subsubsection {* SML *}
haftmann@26182: 
haftmann@26752: code_type Heap (SML "unit/ ->/ _")
haftmann@26182: code_const Heap (SML "raise/ (Fail/ \"bare Heap\")")
haftmann@26753: code_monad run "op \<guillemotright>=" return "()" SML
haftmann@26182: code_const run (SML "_")
haftmann@26752: code_const return (SML "(fn/ ()/ =>/ _)")
haftmann@26182: code_const "Heap_Monad.Fail" (SML "Fail")
haftmann@26752: code_const "Heap_Monad.raise_exc" (SML "(fn/ ()/ =>/ raise/ _)")
haftmann@26182: 
haftmann@26182: 
haftmann@26182: subsubsection {* OCaml *}
haftmann@26182: 
haftmann@26182: code_type Heap (OCaml "_")
haftmann@26182: code_const Heap (OCaml "failwith/ \"bare Heap\"")
haftmann@26753: code_monad run "op \<guillemotright>=" return "()" OCaml
haftmann@26182: code_const run (OCaml "_")
haftmann@27673: code_const return (OCaml "(fun/ ()/ ->/ _)")
haftmann@26182: code_const "Heap_Monad.Fail" (OCaml "Failure")
haftmann@27673: code_const "Heap_Monad.raise_exc" (OCaml "(fun/ ()/ ->/ raise/ _)")
haftmann@26182: 
haftmann@26182: code_reserved OCaml Failure raise
haftmann@26182: 
haftmann@26182: 
haftmann@26182: subsubsection {* Haskell *}
haftmann@26182: 
haftmann@26182: text {* Adaption layer *}
haftmann@26182: 
haftmann@26182: code_include Haskell "STMonad"
haftmann@26182: {*import qualified Control.Monad;
haftmann@26182: import qualified Control.Monad.ST;
haftmann@26182: import qualified Data.STRef;
haftmann@26182: import qualified Data.Array.ST;
haftmann@26182: 
haftmann@27695: type RealWorld = Control.Monad.ST.RealWorld;
haftmann@26182: type ST s a = Control.Monad.ST.ST s a;
haftmann@26182: type STRef s a = Data.STRef.STRef s a;
haftmann@27673: type STArray s a = Data.Array.ST.STArray s Int a;
haftmann@26182: 
haftmann@26182: runST :: (forall s. ST s a) -> a;
haftmann@26182: runST s = Control.Monad.ST.runST s;
haftmann@26182: 
haftmann@26182: newSTRef = Data.STRef.newSTRef;
haftmann@26182: readSTRef = Data.STRef.readSTRef;
haftmann@26182: writeSTRef = Data.STRef.writeSTRef;
haftmann@26182: 
haftmann@27673: newArray :: (Int, Int) -> a -> ST s (STArray s a);
haftmann@26182: newArray = Data.Array.ST.newArray;
haftmann@26182: 
haftmann@27673: newListArray :: (Int, Int) -> [a] -> ST s (STArray s a);
haftmann@26182: newListArray = Data.Array.ST.newListArray;
haftmann@26182: 
haftmann@27673: lengthArray :: STArray s a -> ST s Int;
haftmann@27673: lengthArray a = Control.Monad.liftM snd (Data.Array.ST.getBounds a);
haftmann@26182: 
haftmann@27673: readArray :: STArray s a -> Int -> ST s a;
haftmann@26182: readArray = Data.Array.ST.readArray;
haftmann@26182: 
haftmann@27673: writeArray :: STArray s a -> Int -> a -> ST s ();
haftmann@26182: writeArray = Data.Array.ST.writeArray;*}
haftmann@26182: 
haftmann@27695: code_reserved Haskell RealWorld ST STRef Array
haftmann@26182:   runST
haftmann@26182:   newSTRef reasSTRef writeSTRef
haftmann@27673:   newArray newListArray lengthArray readArray writeArray
haftmann@26182: 
haftmann@26182: text {* Monad *}
haftmann@26182: 
haftmann@27695: code_type Heap (Haskell "ST/ RealWorld/ _")
haftmann@27695: code_const Heap (Haskell "error/ \"bare Heap\"")
haftmann@26752: code_monad run "op \<guillemotright>=" Haskell
haftmann@26182: code_const return (Haskell "return")
haftmann@26182: code_const "Heap_Monad.Fail" (Haskell "_")
haftmann@26182: code_const "Heap_Monad.raise_exc" (Haskell "error")
haftmann@26182: 
haftmann@26170: end




