(* Title: HOL/Library/Heap_Monad.thy ID: $Id$ Author: John Matthews, Galois Connections; Alexander Krauss, Lukas Bulwahn & Florian Haftmann, TU Muenchen *) header {* A monad with a polymorphic heap *} theory Heap_Monad imports Heap begin subsection {* The monad *} subsubsection {* Monad combinators *} datatype exception = Exn text {* Monadic heap actions either produce values and transform the heap, or fail *} datatype 'a Heap = Heap "heap \ ('a + exception) \ heap" primrec execute :: "'a Heap \ heap \ ('a + exception) \ heap" where "execute (Heap f) = f" lemmas [code del] = execute.simps lemma Heap_execute [simp]: "Heap (execute f) = f" by (cases f) simp_all lemma Heap_eqI: "(\h. execute f h = execute g h) \ f = g" by (cases f, cases g) (auto simp: expand_fun_eq) lemma Heap_eqI': "(\h. (\x. execute (f x) h) = (\y. execute (g y) h)) \ f = g" by (auto simp: expand_fun_eq intro: Heap_eqI) lemma Heap_strip: "(\f. PROP P f) \ (\g. PROP P (Heap g))" proof fix g :: "heap \ ('a + exception) \ heap" assume "\f. PROP P f" then show "PROP P (Heap g)" . next fix f :: "'a Heap" assume assm: "\g. PROP P (Heap g)" then have "PROP P (Heap (execute f))" . then show "PROP P f" by simp qed definition heap :: "(heap \ 'a \ heap) \ 'a Heap" where [code del]: "heap f = Heap (\h. apfst Inl (f h))" lemma execute_heap [simp]: "execute (heap f) h = apfst Inl (f h)" by (simp add: heap_def) definition run :: "'a Heap \ 'a Heap" where run_drop [code del]: "run f = f" definition bindM :: "'a Heap \ ('a \ 'b Heap) \ 'b Heap" (infixl ">>=" 54) where [code del]: "f >>= g = Heap (\h. case execute f h of (Inl x, h') \ execute (g x) h' | r \ r)" notation bindM (infixl "\=" 54) abbreviation chainM :: "'a Heap \ 'b Heap \ 'b Heap" (infixl ">>" 54) where "f >> g \ f >>= (\_. g)" notation chainM (infixl "\" 54) definition return :: "'a \ 'a Heap" where [code del]: "return x = heap (Pair x)" lemma execute_return [simp]: "execute (return x) h = apfst Inl (x, h)" by (simp add: return_def) definition raise :: "string \ 'a Heap" where -- {* the string is just decoration *} [code del]: "raise s = Heap (Pair (Inr Exn))" notation (latex output) "raise" ("\<^raw:{\textsf{raise}}>") lemma execute_raise [simp]: "execute (raise s) h = (Inr Exn, h)" by (simp add: raise_def) subsubsection {* do-syntax *} text {* We provide a convenient do-notation for monadic expressions well-known from Haskell. @{const Let} is printed specially in do-expressions. *} nonterminals do_expr syntax "_do" :: "do_expr \ 'a" ("(do (_)//done)" [12] 100) "_bindM" :: "pttrn \ 'a \ do_expr \ do_expr" ("_ <- _;//_" [1000, 13, 12] 12) "_chainM" :: "'a \ do_expr \ do_expr" ("_;//_" [13, 12] 12) "_let" :: "pttrn \ 'a \ do_expr \ do_expr" ("let _ = _;//_" [1000, 13, 12] 12) "_nil" :: "'a \ do_expr" ("_" [12] 12) syntax (xsymbols) "_bindM" :: "pttrn \ 'a \ do_expr \ do_expr" ("_ \ _;//_" [1000, 13, 12] 12) syntax (latex output) "_do" :: "do_expr \ 'a" ("(\<^raw:{\textsf{do}}> (_))" [12] 100) "_let" :: "pttrn \ 'a \ do_expr \ do_expr" ("\<^raw:\textsf{let}> _ = _;//_" [1000, 13, 12] 12) notation (latex output) "return" ("\<^raw:{\textsf{return}}>") translations "_do f" => "CONST run f" "_bindM x f g" => "f \= (\x. g)" "_chainM f g" => "f \ g" "_let x t f" => "CONST Let t (\x. f)" "_nil f" => "f" print_translation {* let fun dest_abs_eta (Abs (abs as (_, ty, _))) = let val (v, t) = Syntax.variant_abs abs; in ((v, ty), t) end | dest_abs_eta t = let val (v, t) = Syntax.variant_abs ("", dummyT, t $ Bound 0); in ((v, dummyT), t) end fun unfold_monad (Const (@{const_syntax bindM}, _) $ f $ g) = let val ((v, ty), g') = dest_abs_eta g; val v_used = fold_aterms (fn Free (w, _) => (fn s => s orelse v = w) | _ => I) g' false; in if v_used then Const ("_bindM", dummyT) $ Free (v, ty) $ f $ unfold_monad g' else Const ("_chainM", dummyT) $ f $ unfold_monad g' end | unfold_monad (Const (@{const_syntax chainM}, _) $ f $ g) = Const ("_chainM", dummyT) $ f $ unfold_monad g | unfold_monad (Const (@{const_syntax Let}, _) $ f $ g) = let val ((v, ty), g') = dest_abs_eta g; in Const ("_let", dummyT) $ Free (v, ty) $ f $ unfold_monad g' end | unfold_monad (Const (@{const_syntax Pair}, _) $ f) = Const ("return", dummyT) $ f | unfold_monad f = f; fun tr' (f::ts) = list_comb (Const ("_do", dummyT) $ unfold_monad f, ts) in [(@{const_syntax "run"}, tr')] end; *} subsection {* Monad properties *} subsubsection {* Superfluous runs *} text {* @{term run} is just a doodle *} lemma run_simp [simp]: "\f. run (run f) = run f" "\f g. run f \= g = f \= g" "\f g. run f \ g = f \ g" "\f g. f \= (\x. run g) = f \= (\x. g)" "\f g. f \ run g = f \ g" "\f. f = run g \ f = g" "\f. run f = g \ f = g" unfolding run_drop by rule+ subsubsection {* Monad laws *} lemma return_bind: "return x \= f = f x" by (simp add: bindM_def return_def) lemma bind_return: "f \= return = f" proof (rule Heap_eqI) fix h show "execute (f \= return) h = execute f h" by (auto simp add: bindM_def return_def split: sum.splits prod.splits) qed lemma bind_bind: "(f \= g) \= h = f \= (\x. g x \= h)" by (rule Heap_eqI) (auto simp add: bindM_def split: split: sum.splits prod.splits) lemma bind_bind': "f \= (\x. g x \= h x) = f \= (\x. g x \= (\y. return (x, y))) \= (\(x, y). h x y)" by (rule Heap_eqI) (auto simp add: bindM_def split: split: sum.splits prod.splits) lemma raise_bind: "raise e \= f = raise e" by (simp add: raise_def bindM_def) lemmas monad_simp = return_bind bind_return bind_bind raise_bind subsection {* Generic combinators *} definition liftM :: "('a \ 'b) \ 'a \ 'b Heap" where "liftM f = return o f" definition compM :: "('a \ 'b Heap) \ ('b \ 'c Heap) \ 'a \ 'c Heap" (infixl ">>==" 54) where "(f >>== g) = (\x. f x \= g)" notation compM (infixl "\==" 54) lemma liftM_collapse: "liftM f x = return (f x)" by (simp add: liftM_def) lemma liftM_compM: "liftM f \== g = g o f" by (auto intro: Heap_eqI' simp add: expand_fun_eq liftM_def compM_def bindM_def) lemma compM_return: "f \== return = f" by (simp add: compM_def monad_simp) lemma compM_compM: "(f \== g) \== h = f \== (g \== h)" by (simp add: compM_def monad_simp) lemma liftM_bind: "(\x. liftM f x \= liftM g) = liftM (\x. g (f x))" by (rule Heap_eqI') (simp add: monad_simp liftM_def bindM_def) lemma liftM_comp: "liftM f o g = liftM (f o g)" by (rule Heap_eqI') (simp add: liftM_def) lemmas monad_simp' = monad_simp liftM_compM compM_return compM_compM liftM_bind liftM_comp primrec mapM :: "('a \ 'b Heap) \ 'a list \ 'b list Heap" where "mapM f [] = return []" | "mapM f (x#xs) = do y \ f x; ys \ mapM f xs; return (y # ys) done" primrec foldM :: "('a \ 'b \ 'b Heap) \ 'a list \ 'b \ 'b Heap" where "foldM f [] s = return s" | "foldM f (x#xs) s = f x s \= foldM f xs" hide (open) const heap execute subsection {* Code generator setup *} subsubsection {* Logical intermediate layer *} definition Fail :: "message_string \ exception" where [code func del]: "Fail s = Exn" definition raise_exc :: "exception \ 'a Heap" where [code func del]: "raise_exc e = raise []" lemma raise_raise_exc [code func, code inline]: "raise s = raise_exc (Fail (STR s))" unfolding Fail_def raise_exc_def raise_def .. hide (open) const Fail raise_exc subsubsection {* SML and OCaml *} code_type Heap (SML "unit/ ->/ _") code_const Heap (SML "raise/ (Fail/ \"bare Heap\")") code_const "op \=" (SML "!(fn/ f/ =>/ fn/ g/ =>/ fn/ ()/ =>/ g/ (f/ ())/ ())") code_const run (SML "_") code_const return (SML "!(fn/ ()/ =>/ _)") code_const "Heap_Monad.Fail" (SML "Fail") code_const "Heap_Monad.raise_exc" (SML "!(fn/ ()/ =>/ raise/ _)") code_type Heap (OCaml "_") code_const Heap (OCaml "failwith/ \"bare Heap\"") code_const "op \=" (OCaml "!(fun/ f/ g/ ()/ ->/ g/ (f/ ())/ ())") code_const run (OCaml "_") code_const return (OCaml "!(fun/ ()/ ->/ _)") code_const "Heap_Monad.Fail" (OCaml "Failure") code_const "Heap_Monad.raise_exc" (OCaml "!(fun/ ()/ ->/ raise/ _)") ML {* local open CodeThingol; val bind' = CodeName.const @{theory} @{const_name bindM}; val return' = CodeName.const @{theory} @{const_name return}; val unit' = CodeName.const @{theory} @{const_name Unity}; fun imp_monad_bind'' ts = let val dummy_name = ""; val dummy_type = ITyVar dummy_name; val dummy_case_term = IVar dummy_name; (*assumption: dummy values are not relevant for serialization*) val unitt = IConst (unit', ([], [])); fun dest_abs ((v, ty) `|-> t, _) = ((v, ty), t) | dest_abs (t, ty) = let val vs = CodeThingol.fold_varnames cons t []; val v = Name.variant vs "x"; val ty' = (hd o fst o CodeThingol.unfold_fun) ty; in ((v, ty'), t `$ IVar v) end; fun force (t as IConst (c, _) `$ t') = if c = return' then t' else t `$ unitt | force t = t `$ unitt; fun tr_bind' [(t1, _), (t2, ty2)] = let val ((v, ty), t) = dest_abs (t2, ty2); in ICase (((force t1, ty), [(IVar v, tr_bind'' t)]), dummy_case_term) end and tr_bind'' t = case CodeThingol.unfold_app t of (IConst (c, (_, ty1 :: ty2 :: _)), [x1, x2]) => if c = bind' then tr_bind' [(x1, ty1), (x2, ty2)] else force t | _ => force t; in (dummy_name, dummy_type) `|-> ICase (((IVar dummy_name, dummy_type), [(unitt, tr_bind' ts)]), dummy_case_term) end and imp_monad_bind' (const as (c, (_, tys))) ts = if c = bind' then case (ts, tys) of ([t1, t2], ty1 :: ty2 :: _) => imp_monad_bind'' [(t1, ty1), (t2, ty2)] | ([t1, t2, t3], ty1 :: ty2 :: _) => imp_monad_bind'' [(t1, ty1), (t2, ty2)] `$ t3 | (ts, _) => imp_monad_bind (eta_expand 2 (const, ts)) else IConst const `$$ map imp_monad_bind ts and imp_monad_bind (IConst const) = imp_monad_bind' const [] | imp_monad_bind (t as IVar _) = t | imp_monad_bind (t as _ `$ _) = (case unfold_app t of (IConst const, ts) => imp_monad_bind' const ts | (t, ts) => imp_monad_bind t `$$ map imp_monad_bind ts) | imp_monad_bind (v_ty `|-> t) = v_ty `|-> imp_monad_bind t | imp_monad_bind (ICase (((t, ty), pats), t0)) = ICase (((imp_monad_bind t, ty), (map o pairself) imp_monad_bind pats), imp_monad_bind t0); in val imp_program = (Graph.map_nodes o map_terms_stmt) imp_monad_bind; end *} setup {* CodeTarget.extend_target ("SML_imp", ("SML", imp_program)) *} setup {* CodeTarget.extend_target ("OCaml_imp", ("OCaml", imp_program)) *} code_reserved OCaml Failure raise subsubsection {* Haskell *} text {* Adaption layer *} code_include Haskell "STMonad" {*import qualified Control.Monad; import qualified Control.Monad.ST; import qualified Data.STRef; import qualified Data.Array.ST; type RealWorld = Control.Monad.ST.RealWorld; type ST s a = Control.Monad.ST.ST s a; type STRef s a = Data.STRef.STRef s a; type STArray s a = Data.Array.ST.STArray s Int a; runST :: (forall s. ST s a) -> a; runST s = Control.Monad.ST.runST s; newSTRef = Data.STRef.newSTRef; readSTRef = Data.STRef.readSTRef; writeSTRef = Data.STRef.writeSTRef; newArray :: (Int, Int) -> a -> ST s (STArray s a); newArray = Data.Array.ST.newArray; newListArray :: (Int, Int) -> [a] -> ST s (STArray s a); newListArray = Data.Array.ST.newListArray; lengthArray :: STArray s a -> ST s Int; lengthArray a = Control.Monad.liftM snd (Data.Array.ST.getBounds a); readArray :: STArray s a -> Int -> ST s a; readArray = Data.Array.ST.readArray; writeArray :: STArray s a -> Int -> a -> ST s (); writeArray = Data.Array.ST.writeArray;*} code_reserved Haskell RealWorld ST STRef Array runST newSTRef reasSTRef writeSTRef newArray newListArray lengthArray readArray writeArray text {* Monad *} code_type Heap (Haskell "ST/ RealWorld/ _") code_const Heap (Haskell "error/ \"bare Heap\"") code_monad run "op \=" Haskell code_const return (Haskell "return") code_const "Heap_Monad.Fail" (Haskell "_") code_const "Heap_Monad.raise_exc" (Haskell "error") end