(* Title: HOL/Library/Heap_Monad.thy ID: $Id: Heap_Monad.thy,v 1.4 2008/04/28 11:41:05 haftmann Exp $ 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. (λx. execute (f x) h) = (λy. 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 (λh. 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 (λh. 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 >>= (λ_. 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" ("{\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" ("_ \<leftarrow> _;//_" [1000, 13, 12] 12) syntax (latex output) "_do" :: "do_expr => 'a" ("({\textsf{do}} (_))" [12] 100) "_let" :: "pttrn => 'a => do_expr => do_expr" ("\textsf{let} _ = _;//_" [1000, 13, 12] 12) notation (latex output) "return" ("{\textsf{return}}") translations "_do f" => "CONST run f" "_bindM x f g" => "f »= (λx. g)" "_chainM f g" => "f » g" "_let x t f" => "CONST Let t (λx. 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; *} subsubsection {* Plain evaluation *} definition evaluate :: "'a Heap => 'a" where [code del]: "evaluate f = (case execute f Heap.empty of (Inl x, _) => x)" 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 »= (λx. run g) = f »= (λx. 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 »= (λx. g x »= h)" by (rule Heap_eqI) (auto simp add: bindM_def split: split: sum.splits prod.splits) lemma bind_bind': "f »= (λx. g x »= h x) = f »= (λx. g x »= (λy. return (x, y))) »= (λ(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) = (λx. 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: "(λx. liftM f x »= liftM g) = liftM (λx. 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 \<leftarrow> f x; ys \<leftarrow> 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 *} code_type Heap (SML "unit/ ->/ _") code_const Heap (SML "raise/ (Fail/ \"bare Heap\")") code_monad run "op »=" return "()" SML 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/ _)") subsubsection {* OCaml *} code_type Heap (OCaml "_") code_const Heap (OCaml "failwith/ \"bare Heap\"") code_monad run "op »=" return "()" OCaml code_const run (OCaml "_") code_const return (OCaml "(fn/ ()/ =>/ _)") code_const "Heap_Monad.Fail" (OCaml "Failure") code_const "Heap_Monad.raise_exc" (OCaml "(fn/ ()/ =>/ raise/ _)") 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 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 Integer 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 :: (Integer, Integer) -> a -> ST s (STArray s a); newArray = Data.Array.ST.newArray; newListArray :: (Integer, Integer) -> [a] -> ST s (STArray s a); newListArray = Data.Array.ST.newListArray; length :: STArray s a -> ST s Integer; length a = Control.Monad.liftM snd (Data.Array.ST.getBounds a); readArray :: STArray s a -> Integer -> ST s a; readArray = Data.Array.ST.readArray; writeArray :: STArray s a -> Integer -> a -> ST s (); writeArray = Data.Array.ST.writeArray;*} code_reserved Haskell ST STRef Array runST newSTRef reasSTRef writeSTRef newArray newListArray bounds readArray writeArray text {* Monad *} code_type Heap (Haskell "ST '_s _") code_const Heap (Haskell "error \"bare Heap\")") code_const evaluate (Haskell "runST") 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
lemma
execute (Heap f) = f
lemma Heap_execute:
Heap (execute f) = f
lemma Heap_eqI:
(!!h. execute f h = execute g h) ==> f = g
lemma Heap_eqI':
(!!h. (λx. execute (f x) h) = (λy. execute (g y) h)) ==> f = g
lemma Heap_strip:
(!!f. PROP P f) == (!!g. PROP P (Heap g))
lemma execute_heap:
execute (Heap_Monad.heap f) h = apfst Inl (f h)
lemma execute_return:
execute (return x) h = apfst Inl (x, h)
lemma execute_raise:
execute (raise s) h = (Inr Exn, h)
lemma run_simp:
do do f
done
done =
do f
done
do f
done »=
g =
f »= g
do f
done »
g =
f » g
f » do g
done =
f » g
f » do g
done =
f » g
(f = do g
done) =
(f = g)
(do f
done =
g) =
(f = g)
lemma return_bind:
return x »= f = f x
lemma bind_return:
f »= return = f
lemma bind_bind:
f »= g »= h = f »= (λx. g x »= h)
lemma bind_bind':
f »= (λx. g x »= h x) =
f »= (λx. g x »= (λy. return (x, y))) »= (λ(x, y). h x y)
lemma raise_bind:
raise e »= f = raise e
lemma monad_simp:
return x »= f = f x
f »= return = f
f »= g »= h = f »= (λx. g x »= h)
raise e »= f = raise e
lemma liftM_collapse:
liftM f x = return (f x)
lemma liftM_compM:
liftM f »== g = g o f
lemma compM_return:
f »== return = f
lemma compM_compM:
f »== g »== h = f »== (g »== h)
lemma liftM_bind:
(λx. liftM f x »= liftM g) = liftM (λx. g (f x))
lemma liftM_comp:
liftM f o g = liftM (f o g)
lemma monad_simp':
return x »= f = f x
f »= return = f
f »= g »= h = f »= (λx. g x »= h)
raise e »= f = raise e
liftM f »== g = g o f
f »== return = f
f »== g »== h = f »== (g »== h)
(λx. liftM f x »= liftM g) = liftM (λx. g (f x))
liftM f o g = liftM (f o g)
lemma raise_raise_exc:
raise s = raise_exc (Fail (STR s))