(*  Title:      HOL/Library/Heap_Monad.thy

    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 (@{syntax_const "_bindM"}, dummyT) $ v $ f $ unfold_monad g'

        else

          Const (@{syntax_const "_chainM"}, dummyT) $ f $ unfold_monad g'

        end

    | unfold_monad (Const (@{const_syntax chainM}, _) $ f $ g) =

        Const (@{syntax_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 (@{syntax_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 (@{syntax_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 (@{syntax_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)

subsubsection {* A monadic combinator for simple recursive functions *}

function (default "\<lambda>(f,g,x,h). (Inr Exn, undefined)") 

  mrec 

where

  "mrec f g x h = 

   (case Heap_Monad.execute (f x) h of

     (Inl (Inl r), h') \<Rightarrow> (Inl r, h')

   | (Inl (Inr s), h') \<Rightarrow> 

          (case mrec f g s h' of

             (Inl z, h'') \<Rightarrow> Heap_Monad.execute (g x s z) h''

           | (Inr e, h'') \<Rightarrow> (Inr e, h''))

   | (Inr e, h') \<Rightarrow> (Inr e, h')

   )"

by auto

lemma graph_implies_dom:

	"mrec_graph x y \<Longrightarrow> mrec_dom x"

apply (induct rule:mrec_graph.induct) 

apply (rule accpI)

apply (erule mrec_rel.cases)

by simp

lemma f_default: "\<not> mrec_dom (f, g, x, h) \<Longrightarrow> mrec f g x h = (Inr Exn, undefined)"

	unfolding mrec_def 

  by (rule fundef_default_value[OF mrec_sumC_def graph_implies_dom, of _ _ "(f, g, x, h)", simplified])

lemma f_di_reverse: 

  assumes "\<not> mrec_dom (f, g, x, h)"

  shows "

   (case Heap_Monad.execute (f x) h of

     (Inl (Inl r), h') \<Rightarrow> mrecalse

   | (Inl (Inr s), h') \<Rightarrow> \<not> mrec_dom (f, g, s, h')

   | (Inr e, h') \<Rightarrow> mrecalse

   )" 

using assms

by (auto split:prod.splits sum.splits)

 (erule notE, rule accpI, elim mrec_rel.cases, simp)+

lemma mrec_rule:

  "mrec f g x h = 

   (case Heap_Monad.execute (f x) h of

     (Inl (Inl r), h') \<Rightarrow> (Inl r, h')

   | (Inl (Inr s), h') \<Rightarrow> 

          (case mrec f g s h' of

             (Inl z, h'') \<Rightarrow> Heap_Monad.execute (g x s z) h''

           | (Inr e, h'') \<Rightarrow> (Inr e, h''))

   | (Inr e, h') \<Rightarrow> (Inr e, h')

   )"

apply (cases "mrec_dom (f,g,x,h)", simp)

apply (frule f_default)

apply (frule f_di_reverse, simp)

by (auto split: sum.split prod.split simp: f_default)

definition

  "MREC f g x = Heap (mrec f g x)"

lemma MREC_rule:

  "MREC f g x = 

  (do y \<leftarrow> f x;

                (case y of 

                Inl r \<Rightarrow> return r

              | Inr s \<Rightarrow> 

                do z \<leftarrow> MREC f g s ;

                   g x s z

                done) done)"

  unfolding MREC_def

  unfolding bindM_def return_def

  apply simp

  apply (rule ext)

  apply (unfold mrec_rule[of f g x])

  by (auto split:prod.splits sum.splits)

hide (open) const heap execute

subsection {* Code generator setup *}

subsubsection {* Logical intermediate layer *}

definition

  Fail :: "String.literal \<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_unfold]:

  "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 imp_program naming =

  let

    fun is_const c = case lookup_const naming c

     of SOME c' => (fn c'' => c' = c'')

      | NONE => K false;

    val is_bindM = is_const @{const_name bindM};

    val is_return = is_const @{const_name return};

    val dummy_name = "";

    val dummy_type = ITyVar dummy_name;

    val dummy_case_term = IVar NONE;

    (*assumption: dummy values are not relevant for serialization*)

    val unitt = case lookup_const naming @{const_name Unity}

     of SOME unit' => IConst (unit', (([], []), []))

      | NONE => error ("Must include " ^ @{const_name Unity} ^ " in generated constants.");

    fun dest_abs ((v, ty) `|=> t, _) = ((v, ty), t)

      | dest_abs (t, ty) =

          let

            val vs = fold_varnames cons t [];

            val v = Name.variant vs "x";

            val ty' = (hd o fst o unfold_fun) ty;

          in ((SOME v, ty'), t `$ IVar (SOME v)) end;

    fun force (t as IConst (c, _) `$ t') = if is_return c

          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 unfold_app t

         of (IConst (c, (_, ty1 :: ty2 :: _)), [x1, x2]) => if is_bindM c

              then tr_bind' [(x1, ty1), (x2, ty2)]

              else force t

          | _ => force t;

    fun imp_monad_bind'' ts = (SOME dummy_name, dummy_type) `|=> ICase (((IVar (SOME dummy_name), dummy_type),

      [(unitt, tr_bind' ts)]), dummy_case_term)

    and imp_monad_bind' (const as (c, (_, tys))) ts = if is_bindM c 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 (Graph.map_nodes o map_terms_stmt) imp_monad_bind end;

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 "Heap"

{*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;

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 Heap

text {* Monad *}

code_type Heap (Haskell "Heap.ST/ Heap.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