(* 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 \<Rightarrow> ('a + exception) \<times> heap"
primrec
execute :: "'a Heap \<Rightarrow> heap \<Rightarrow> ('a + exception) \<times> 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:
"(\<And>h. execute f h = execute g h) \<Longrightarrow> f = g"
by (cases f, cases g) (auto simp: expand_fun_eq)
lemma Heap_eqI':
"(\<And>h. (\<lambda>x. execute (f x) h) = (\<lambda>y. execute (g y) h)) \<Longrightarrow> f = g"
by (auto simp: expand_fun_eq intro: Heap_eqI)
lemma Heap_strip: "(\<And>f. PROP P f) \<equiv> (\<And>g. PROP P (Heap g))"
proof
fix g :: "heap \<Rightarrow> ('a + exception) \<times> heap"
assume "\<And>f. PROP P f"
then show "PROP P (Heap g)" .
next
fix f :: "'a Heap"
assume assm: "\<And>g. PROP P (Heap g)"
then have "PROP P (Heap (execute f))" .
then show "PROP P f" by simp
qed
definition
heap :: "(heap \<Rightarrow> 'a \<times> heap) \<Rightarrow> 'a Heap" where
[code del]: "heap f = Heap (\<lambda>h. apfst Inl (f h))"
lemma execute_heap [simp]:
"execute (heap f) h = apfst Inl (f h)"
by (simp add: heap_def)
definition
bindM :: "'a Heap \<Rightarrow> ('a \<Rightarrow> 'b Heap) \<Rightarrow> 'b Heap" (infixl ">>=" 54) where
[code del]: "f >>= g = Heap (\<lambda>h. case execute f h of
(Inl x, h') \<Rightarrow> execute (g x) h'
| r \<Rightarrow> r)"
notation
bindM (infixl "\<guillemotright>=" 54)
abbreviation
chainM :: "'a Heap \<Rightarrow> 'b Heap \<Rightarrow> 'b Heap" (infixl ">>" 54) where
"f >> g \<equiv> f >>= (\<lambda>_. g)"
notation
chainM (infixl "\<guillemotright>" 54)
definition
return :: "'a \<Rightarrow> '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 \<Rightarrow> '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 \<Rightarrow> 'a"
("(do (_)//done)" [12] 100)
"_bindM" :: "pttrn \<Rightarrow> 'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
("_ <- _;//_" [1000, 13, 12] 12)
"_chainM" :: "'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
("_;//_" [13, 12] 12)
"_let" :: "pttrn \<Rightarrow> 'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
("let _ = _;//_" [1000, 13, 12] 12)
"_nil" :: "'a \<Rightarrow> do_expr"
("_" [12] 12)
syntax (xsymbols)
"_bindM" :: "pttrn \<Rightarrow> 'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
("_ \<leftarrow> _;//_" [1000, 13, 12] 12)
syntax (latex output)
"_do" :: "do_expr \<Rightarrow> 'a"
("(\<^raw:{\textsf{do}}> (_))" [12] 100)
"_let" :: "pttrn \<Rightarrow> 'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
("\<^raw:\textsf{let}> _ = _;//_" [1000, 13, 12] 12)
notation (latex output)
"return" ("\<^raw:{\textsf{return}}>")
translations
"_do f" => "f"
"_bindM x f g" => "f \<guillemotright>= (\<lambda>x. g)"
"_chainM f g" => "f \<guillemotright> g"
"_let x t f" => "CONST Let t (\<lambda>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 (Free (v, ty), t) end
| dest_abs_eta t =
let
val (v, t) = Syntax.variant_abs ("", dummyT, t $ Bound 0);
in (Free (v, dummyT), t) end;
fun unfold_monad (Const (@{const_syntax bindM}, _) $ f $ g) =
let
val (v, g') = dest_abs_eta g;
val vs = fold_aterms (fn Free (v, _) => insert (op =) v | _ => I) v [];
val v_used = fold_aterms
(fn Free (w, _) => (fn s => s orelse member (op =) vs w) | _ => I) g' false;
in if v_used then
Const ("_bindM", dummyT) $ v $ 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, g') = dest_abs_eta g;
in Const ("_let", dummyT) $ v $ f $ unfold_monad g' end
| unfold_monad (Const (@{const_syntax Pair}, _) $ f) =
Const (@{const_syntax return}, dummyT) $ f
| unfold_monad f = f;
fun contains_bindM (Const (@{const_syntax bindM}, _) $ _ $ _) = true
| contains_bindM (Const (@{const_syntax Let}, _) $ _ $ Abs (_, _, t)) =
contains_bindM t;
fun bindM_monad_tr' (f::g::ts) = list_comb
(Const ("_do", dummyT) $ unfold_monad (Const (@{const_syntax bindM}, dummyT) $ f $ g), ts);
fun Let_monad_tr' (f :: (g as Abs (_, _, g')) :: ts) = if contains_bindM g' then list_comb
(Const ("_do", dummyT) $ unfold_monad (Const (@{const_syntax Let}, dummyT) $ f $ g), ts)
else raise Match;
in [
(@{const_syntax bindM}, bindM_monad_tr'),
(@{const_syntax Let}, Let_monad_tr')
] end;
*}
subsection {* Monad properties *}
subsubsection {* Monad laws *}
lemma return_bind: "return x \<guillemotright>= f = f x"
by (simp add: bindM_def return_def)
lemma bind_return: "f \<guillemotright>= return = f"
proof (rule Heap_eqI)
fix h
show "execute (f \<guillemotright>= return) h = execute f h"
by (auto simp add: bindM_def return_def split: sum.splits prod.splits)
qed
lemma bind_bind: "(f \<guillemotright>= g) \<guillemotright>= h = f \<guillemotright>= (\<lambda>x. g x \<guillemotright>= h)"
by (rule Heap_eqI) (auto simp add: bindM_def split: split: sum.splits prod.splits)
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)"
by (rule Heap_eqI) (auto simp add: bindM_def split: split: sum.splits prod.splits)
lemma raise_bind: "raise e \<guillemotright>= 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 \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b Heap"
where
"liftM f = return o f"
definition
compM :: "('a \<Rightarrow> 'b Heap) \<Rightarrow> ('b \<Rightarrow> 'c Heap) \<Rightarrow> 'a \<Rightarrow> 'c Heap" (infixl ">>==" 54)
where
"(f >>== g) = (\<lambda>x. f x \<guillemotright>= g)"
notation
compM (infixl "\<guillemotright>==" 54)
lemma liftM_collapse: "liftM f x = return (f x)"
by (simp add: liftM_def)
lemma liftM_compM: "liftM f \<guillemotright>== g = g o f"
by (auto intro: Heap_eqI' simp add: expand_fun_eq liftM_def compM_def bindM_def)
lemma compM_return: "f \<guillemotright>== return = f"
by (simp add: compM_def monad_simp)
lemma compM_compM: "(f \<guillemotright>== g) \<guillemotright>== h = f \<guillemotright>== (g \<guillemotright>== h)"
by (simp add: compM_def monad_simp)
lemma liftM_bind:
"(\<lambda>x. liftM f x \<guillemotright>= liftM g) = liftM (\<lambda>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 \<Rightarrow> 'b Heap) \<Rightarrow> 'a list \<Rightarrow> '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 \<Rightarrow> 'b \<Rightarrow> 'b Heap) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b Heap"
where
"foldM f [] s = return s"
| "foldM f (x#xs) s = f x s \<guillemotright>= foldM f xs"
definition
assert :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a Heap"
where
"assert P x = (if P x then return x else raise (''assert''))"
lemma assert_cong [fundef_cong]:
assumes "P = P'"
assumes "\<And>x. P' x \<Longrightarrow> f x = f' x"
shows "(assert P x >>= f) = (assert P' x >>= f')"
using assms by (auto simp add: assert_def return_bind raise_bind)
hide (open) const heap execute
subsection {* Code generator setup *}
subsubsection {* Logical intermediate layer *}
definition
Fail :: "message_string \<Rightarrow> exception"
where
[code del]: "Fail s = Exn"
definition
raise_exc :: "exception \<Rightarrow> 'a Heap"
where
[code del]: "raise_exc e = raise []"
lemma raise_raise_exc [code, 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 \<guillemotright>=" (SML "!(fn/ f'_/ =>/ fn/ ()/ =>/ f'_/ (_/ ())/ ())")
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 \<guillemotright>=" (OCaml "!(fun/ f'_/ ()/ ->/ f'_/ (_/ ())/ ())")
code_const return (OCaml "!(fun/ ()/ ->/ _)")
code_const "Heap_Monad.Fail" (OCaml "Failure")
code_const "Heap_Monad.raise_exc" (OCaml "!(fun/ ()/ ->/ raise/ _)")
setup {* let
open Code_Thingol;
fun lookup naming = the o Code_Thingol.lookup_const naming;
fun imp_monad_bind'' bind' return' unit' 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 = Code_Thingol.fold_varnames cons t [];
val v = Name.variant vs "x";
val ty' = (hd o fst o Code_Thingol.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 Code_Thingol.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' bind' return' unit' (const as (c, (_, tys))) ts = if c = bind' then case (ts, tys)
of ([t1, t2], ty1 :: ty2 :: _) => imp_monad_bind'' bind' return' unit' [(t1, ty1), (t2, ty2)]
| ([t1, t2, t3], ty1 :: ty2 :: _) => imp_monad_bind'' bind' return' unit' [(t1, ty1), (t2, ty2)] `$ t3
| (ts, _) => imp_monad_bind bind' return' unit' (eta_expand 2 (const, ts))
else IConst const `$$ map (imp_monad_bind bind' return' unit') ts
and imp_monad_bind bind' return' unit' (IConst const) = imp_monad_bind' bind' return' unit' const []
| imp_monad_bind bind' return' unit' (t as IVar _) = t
| imp_monad_bind bind' return' unit' (t as _ `$ _) = (case unfold_app t
of (IConst const, ts) => imp_monad_bind' bind' return' unit' const ts
| (t, ts) => imp_monad_bind bind' return' unit' t `$$ map (imp_monad_bind bind' return' unit') ts)
| imp_monad_bind bind' return' unit' (v_ty `|-> t) = v_ty `|-> imp_monad_bind bind' return' unit' t
| imp_monad_bind bind' return' unit' (ICase (((t, ty), pats), t0)) = ICase
(((imp_monad_bind bind' return' unit' t, ty), (map o pairself) (imp_monad_bind bind' return' unit') pats), imp_monad_bind bind' return' unit' t0);
fun imp_program naming = (Graph.map_nodes o map_terms_stmt)
(imp_monad_bind (lookup naming @{const_name bindM})
(lookup naming @{const_name return})
(lookup naming @{const_name Unity}));
in
Code_Target.extend_target ("SML_imp", ("SML", imp_program))
#> Code_Target.extend_target ("OCaml_imp", ("OCaml", imp_program))
end
*}
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 "op \<guillemotright>=" Haskell
code_const return (Haskell "return")
code_const "Heap_Monad.Fail" (Haskell "_")
code_const "Heap_Monad.raise_exc" (Haskell "error")
end