(*  Title:      HOL/Imperative_HOL/Heap_Monad.thy

     Author:     John Matthews, Galois Connections; Alexander Krauss, Lukas Bulwahn & Florian Haftmann, TU Muenchen

 *)

 header {* A monad with a polymorphic heap and primitive reasoning infrastructure *}

 theory Heap_Monad

 imports Heap Monad_Syntax Code_Natural

 begin

 subsection {* The monad *}

 subsubsection {* Monad construction *}

 text {* Monadic heap actions either produce values

   and transform the heap, or fail *}

 datatype 'a Heap = Heap "heap \<Rightarrow> ('a \<times> heap) option"

 primrec execute :: "'a Heap \<Rightarrow> heap \<Rightarrow> ('a \<times> heap) option" where

   [code del]: "execute (Heap f) = f"

 lemma Heap_cases [case_names succeed fail]:

   fixes f and h

   assumes succeed: "\<And>x h'. execute f h = Some (x, h') \<Longrightarrow> P"

   assumes fail: "execute f h = None \<Longrightarrow> P"

   shows P

   using assms by (cases "execute f h") auto

 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)

 ML {* structure Execute_Simps = Named_Thms(

   val name = "execute_simps"

   val description = "simplification rules for execute"

 ) *}

 setup Execute_Simps.setup

 lemma execute_Let [execute_simps]:

   "execute (let x = t in f x) = (let x = t in execute (f x))"

   by (simp add: Let_def)

 subsubsection {* Specialised lifters *}

 definition tap :: "(heap \<Rightarrow> 'a) \<Rightarrow> 'a Heap" where

   [code del]: "tap f = Heap (\<lambda>h. Some (f h, h))"

 lemma execute_tap [execute_simps]:

   "execute (tap f) h = Some (f h, h)"

   by (simp add: tap_def)

 definition heap :: "(heap \<Rightarrow> 'a \<times> heap) \<Rightarrow> 'a Heap" where

   [code del]: "heap f = Heap (Some \<circ> f)"

 lemma execute_heap [execute_simps]:

   "execute (heap f) = Some \<circ> f"

   by (simp add: heap_def)

 definition guard :: "(heap \<Rightarrow> bool) \<Rightarrow> (heap \<Rightarrow> 'a \<times> heap) \<Rightarrow> 'a Heap" where

   [code del]: "guard P f = Heap (\<lambda>h. if P h then Some (f h) else None)"

 lemma execute_guard [execute_simps]:

   "\<not> P h \<Longrightarrow> execute (guard P f) h = None"

   "P h \<Longrightarrow> execute (guard P f) h = Some (f h)"

   by (simp_all add: guard_def)

 subsubsection {* Predicate classifying successful computations *}

 definition success :: "'a Heap \<Rightarrow> heap \<Rightarrow> bool" where

   "success f h \<longleftrightarrow> execute f h \<noteq> None"

 lemma successI:

   "execute f h \<noteq> None \<Longrightarrow> success f h"

   by (simp add: success_def)

 lemma successE:

   assumes "success f h"

   obtains r h' where "r = fst (the (execute c h))"

     and "h' = snd (the (execute c h))"

     and "execute f h \<noteq> None"

   using assms by (simp add: success_def)

 ML {* structure Success_Intros = Named_Thms(

   val name = "success_intros"

   val description = "introduction rules for success"

 ) *}

 setup Success_Intros.setup

 lemma success_tapI [success_intros]:

   "success (tap f) h"

   by (rule successI) (simp add: execute_simps)

 lemma success_heapI [success_intros]:

   "success (heap f) h"

   by (rule successI) (simp add: execute_simps)

 lemma success_guardI [success_intros]:

   "P h \<Longrightarrow> success (guard P f) h"

   by (rule successI) (simp add: execute_guard)

 lemma success_LetI [success_intros]:

   "x = t \<Longrightarrow> success (f x) h \<Longrightarrow> success (let x = t in f x) h"

   by (simp add: Let_def)

 lemma success_ifI:

   "(c \<Longrightarrow> success t h) \<Longrightarrow> (\<not> c \<Longrightarrow> success e h) \<Longrightarrow>

     success (if c then t else e) h"

   by (simp add: success_def)

 subsubsection {* Predicate for a simple relational calculus *}

 text {*

   The @{text crel} predicate states that when a computation @{text c}

   runs with the heap @{text h} will result in return value @{text r}

   and a heap @{text "h'"}, i.e.~no exception occurs.

 *}  

 definition crel :: "'a Heap \<Rightarrow> heap \<Rightarrow> heap \<Rightarrow> 'a \<Rightarrow> bool" where

   crel_def: "crel c h h' r \<longleftrightarrow> execute c h = Some (r, h')"

 lemma crelI:

   "execute c h = Some (r, h') \<Longrightarrow> crel c h h' r"

   by (simp add: crel_def)

 lemma crelE:

   assumes "crel c h h' r"

   obtains "r = fst (the (execute c h))"

     and "h' = snd (the (execute c h))"

     and "success c h"

 proof (rule that)

   from assms have *: "execute c h = Some (r, h')" by (simp add: crel_def)

   then show "success c h" by (simp add: success_def)

   from * have "fst (the (execute c h)) = r" and "snd (the (execute c h)) = h'"

     by simp_all

   then show "r = fst (the (execute c h))"

     and "h' = snd (the (execute c h))" by simp_all

 qed

 lemma crel_success:

   "crel c h h' r \<Longrightarrow> success c h"

   by (simp add: crel_def success_def)

 lemma success_crelE:

   assumes "success c h"

   obtains r h' where "crel c h h' r"

   using assms by (auto simp add: crel_def success_def)

 lemma crel_deterministic:

   assumes "crel f h h' a"

     and "crel f h h'' b"

   shows "a = b" and "h' = h''"

   using assms unfolding crel_def by auto

 ML {* structure Crel_Intros = Named_Thms(

   val name = "crel_intros"

   val description = "introduction rules for crel"

 ) *}

 ML {* structure Crel_Elims = Named_Thms(

   val name = "crel_elims"

   val description = "elimination rules for crel"

 ) *}

 setup "Crel_Intros.setup #> Crel_Elims.setup"

 lemma crel_LetI [crel_intros]:

   assumes "x = t" "crel (f x) h h' r"

   shows "crel (let x = t in f x) h h' r"

   using assms by simp

 lemma crel_LetE [crel_elims]:

   assumes "crel (let x = t in f x) h h' r"

   obtains "crel (f t) h h' r"

   using assms by simp

 lemma crel_ifI:

   assumes "c \<Longrightarrow> crel t h h' r"

     and "\<not> c \<Longrightarrow> crel e h h' r"

   shows "crel (if c then t else e) h h' r"

   by (cases c) (simp_all add: assms)

 lemma crel_ifE:

   assumes "crel (if c then t else e) h h' r"

   obtains "c" "crel t h h' r"

     | "\<not> c" "crel e h h' r"

   using assms by (cases c) simp_all

 lemma crel_tapI [crel_intros]:

   assumes "h' = h" "r = f h"

   shows "crel (tap f) h h' r"

   by (rule crelI) (simp add: assms execute_simps)

 lemma crel_tapE [crel_elims]:

   assumes "crel (tap f) h h' r"

   obtains "h' = h" and "r = f h"

   using assms by (rule crelE) (auto simp add: execute_simps)

 lemma crel_heapI [crel_intros]:

   assumes "h' = snd (f h)" "r = fst (f h)"

   shows "crel (heap f) h h' r"

   by (rule crelI) (simp add: assms execute_simps)

 lemma crel_heapE [crel_elims]:

   assumes "crel (heap f) h h' r"

   obtains "h' = snd (f h)" and "r = fst (f h)"

   using assms by (rule crelE) (simp add: execute_simps)

 lemma crel_guardI [crel_intros]:

   assumes "P h" "h' = snd (f h)" "r = fst (f h)"

   shows "crel (guard P f) h h' r"

   by (rule crelI) (simp add: assms execute_simps)

 lemma crel_guardE [crel_elims]:

   assumes "crel (guard P f) h h' r"

   obtains "h' = snd (f h)" "r = fst (f h)" "P h"

   using assms by (rule crelE)

     (auto simp add: execute_simps elim!: successE, cases "P h", auto simp add: execute_simps)

 subsubsection {* Monad combinators *}

 definition return :: "'a \<Rightarrow> 'a Heap" where

   [code del]: "return x = heap (Pair x)"

 lemma execute_return [execute_simps]:

   "execute (return x) = Some \<circ> Pair x"

   by (simp add: return_def execute_simps)

 lemma success_returnI [success_intros]:

   "success (return x) h"

   by (rule successI) (simp add: execute_simps)

 lemma crel_returnI [crel_intros]:

   "h = h' \<Longrightarrow> crel (return x) h h' x"

   by (rule crelI) (simp add: execute_simps)

 lemma crel_returnE [crel_elims]:

   assumes "crel (return x) h h' r"

   obtains "r = x" "h' = h"

   using assms by (rule crelE) (simp add: execute_simps)

 definition raise :: "string \<Rightarrow> 'a Heap" where -- {* the string is just decoration *}

   [code del]: "raise s = Heap (\<lambda>_. None)"

 lemma execute_raise [execute_simps]:

   "execute (raise s) = (\<lambda>_. None)"

   by (simp add: raise_def)

 lemma crel_raiseE [crel_elims]:

   assumes "crel (raise x) h h' r"

   obtains "False"

   using assms by (rule crelE) (simp add: success_def execute_simps)

 definition bind :: "'a Heap \<Rightarrow> ('a \<Rightarrow> 'b Heap) \<Rightarrow> 'b Heap" where

   [code del]: "bind f g = Heap (\<lambda>h. case execute f h of

                   Some (x, h') \<Rightarrow> execute (g x) h'

                 | None \<Rightarrow> None)"

 setup {*

   Adhoc_Overloading.add_variant 

     @{const_name Monad_Syntax.bind} @{const_name Heap_Monad.bind}

 *}

 lemma execute_bind [execute_simps]:

   "execute f h = Some (x, h') \<Longrightarrow> execute (f \<guillemotright>= g) h = execute (g x) h'"

   "execute f h = None \<Longrightarrow> execute (f \<guillemotright>= g) h = None"

   by (simp_all add: bind_def)

 lemma execute_bind_case:

   "execute (f \<guillemotright>= g) h = (case (execute f h) of

     Some (x, h') \<Rightarrow> execute (g x) h' | None \<Rightarrow> None)"

   by (simp add: bind_def)

 lemma execute_bind_success:

   "success f h \<Longrightarrow> execute (f \<guillemotright>= g) h = execute (g (fst (the (execute f h)))) (snd (the (execute f h)))"

   by (cases f h rule: Heap_cases) (auto elim!: successE simp add: bind_def)

 lemma success_bind_executeI:

   "execute f h = Some (x, h') \<Longrightarrow> success (g x) h' \<Longrightarrow> success (f \<guillemotright>= g) h"

   by (auto intro!: successI elim!: successE simp add: bind_def)

 lemma success_bind_crelI [success_intros]:

   "crel f h h' x \<Longrightarrow> success (g x) h' \<Longrightarrow> success (f \<guillemotright>= g) h"

   by (auto simp add: crel_def success_def bind_def)

 lemma crel_bindI [crel_intros]:

   assumes "crel f h h' r" "crel (g r) h' h'' r'"

   shows "crel (f \<guillemotright>= g) h h'' r'"

   using assms

   apply (auto intro!: crelI elim!: crelE successE)

   apply (subst execute_bind, simp_all)

   done

 lemma crel_bindE [crel_elims]:

   assumes "crel (f \<guillemotright>= g) h h'' r'"

   obtains h' r where "crel f h h' r" "crel (g r) h' h'' r'"

   using assms by (auto simp add: crel_def bind_def split: option.split_asm)

 lemma execute_bind_eq_SomeI:

   assumes "execute f h = Some (x, h')"

     and "execute (g x) h' = Some (y, h'')"

   shows "execute (f \<guillemotright>= g) h = Some (y, h'')"

   using assms by (simp add: bind_def)

 lemma return_bind [simp]: "return x \<guillemotright>= f = f x"

   by (rule Heap_eqI) (simp add: execute_bind execute_simps)

 lemma bind_return [simp]: "f \<guillemotright>= return = f"

   by (rule Heap_eqI) (simp add: bind_def execute_simps split: option.splits)

 lemma bind_bind [simp]: "(f \<guillemotright>= g) \<guillemotright>= k = (f :: 'a Heap) \<guillemotright>= (\<lambda>x. g x \<guillemotright>= k)"

   by (rule Heap_eqI) (simp add: bind_def execute_simps split: option.splits)

 lemma raise_bind [simp]: "raise e \<guillemotright>= f = raise e"

   by (rule Heap_eqI) (simp add: execute_simps)

 subsection {* Generic combinators *}

 subsubsection {* Assertions *}

 definition assert :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a Heap" where

   "assert P x = (if P x then return x else raise ''assert'')"

 lemma execute_assert [execute_simps]:

   "P x \<Longrightarrow> execute (assert P x) h = Some (x, h)"

   "\<not> P x \<Longrightarrow> execute (assert P x) h = None"

   by (simp_all add: assert_def execute_simps)

 lemma success_assertI [success_intros]:

   "P x \<Longrightarrow> success (assert P x) h"

   by (rule successI) (simp add: execute_assert)

 lemma crel_assertI [crel_intros]:

   "P x \<Longrightarrow> h' = h \<Longrightarrow> r = x \<Longrightarrow> crel (assert P x) h h' r"

   by (rule crelI) (simp add: execute_assert)

 lemma crel_assertE [crel_elims]:

   assumes "crel (assert P x) h h' r"

   obtains "P x" "r = x" "h' = h"

   using assms by (rule crelE) (cases "P x", simp_all add: execute_assert success_def)

 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')"

   by (rule Heap_eqI) (insert assms, simp add: assert_def)

 subsubsection {* Plain lifting *}

 definition lift :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b Heap" where

   "lift f = return o f"

 lemma lift_collapse [simp]:

   "lift f x = return (f x)"

   by (simp add: lift_def)

 lemma bind_lift:

   "(f \<guillemotright>= lift g) = (f \<guillemotright>= (\<lambda>x. return (g x)))"

   by (simp add: lift_def comp_def)

 subsubsection {* Iteration -- warning: this is rarely useful! *}

 primrec fold_map :: "('a \<Rightarrow> 'b Heap) \<Rightarrow> 'a list \<Rightarrow> 'b list Heap" where

   "fold_map f [] = return []"

 | "fold_map f (x # xs) = do {

      y \<leftarrow> f x;

      ys \<leftarrow> fold_map f xs;

      return (y # ys)

    }"

 lemma fold_map_append:

   "fold_map f (xs @ ys) = fold_map f xs \<guillemotright>= (\<lambda>xs. fold_map f ys \<guillemotright>= (\<lambda>ys. return (xs @ ys)))"

   by (induct xs) simp_all

 lemma execute_fold_map_unchanged_heap [execute_simps]:

   assumes "\<And>x. x \<in> set xs \<Longrightarrow> \<exists>y. execute (f x) h = Some (y, h)"

   shows "execute (fold_map f xs) h =

     Some (List.map (\<lambda>x. fst (the (execute (f x) h))) xs, h)"

 using assms proof (induct xs)

   case Nil show ?case by (simp add: execute_simps)

 next

   case (Cons x xs)

   from Cons.prems obtain y

     where y: "execute (f x) h = Some (y, h)" by auto

   moreover from Cons.prems Cons.hyps have "execute (fold_map f xs) h =

     Some (map (\<lambda>x. fst (the (execute (f x) h))) xs, h)" by auto

   ultimately show ?case by (simp, simp only: execute_bind(1), simp add: execute_simps)

 qed

 subsection {* Code generator setup *}

 subsubsection {* Logical intermediate layer *}

 primrec raise' :: "String.literal \<Rightarrow> 'a Heap" where

   [code del, code_post]: "raise' (STR s) = raise s"

 lemma raise_raise' [code_inline]:

   "raise s = raise' (STR s)"

   by simp

 code_datatype raise' -- {* avoid @{const "Heap"} formally *}

 subsubsection {* SML and OCaml *}

 code_type Heap (SML "unit/ ->/ _")

 code_const bind (SML "!(fn/ f'_/ =>/ fn/ ()/ =>/ f'_/ (_/ ())/ ())")

 code_const return (SML "!(fn/ ()/ =>/ _)")

 code_const Heap_Monad.raise' (SML "!(raise/ Fail/ _)")

 code_type Heap (OCaml "unit/ ->/ _")

 code_const bind (OCaml "!(fun/ f'_/ ()/ ->/ f'_/ (_/ ())/ ())")

 code_const return (OCaml "!(fun/ ()/ ->/ _)")

 code_const Heap_Monad.raise' (OCaml "failwith")

 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;

 import Natural;

 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 Natural a;

 newSTRef = Data.STRef.newSTRef;

 readSTRef = Data.STRef.readSTRef;

 writeSTRef = Data.STRef.writeSTRef;

 newArray :: Natural -> a -> ST s (STArray s a);

 newArray k = Data.Array.ST.newArray (0, k);

 newListArray :: [a] -> ST s (STArray s a);

 newListArray xs = Data.Array.ST.newListArray (0, (fromInteger . toInteger . length) xs) xs;

 newFunArray :: Natural -> (Natural -> a) -> ST s (STArray s a);

 newFunArray k f = Data.Array.ST.newListArray (0, k) (map f [0..k-1]);

 lengthArray :: STArray s a -> ST s Natural;

 lengthArray a = Control.Monad.liftM snd (Data.Array.ST.getBounds a);

 readArray :: STArray s a -> Natural -> ST s a;

 readArray = Data.Array.ST.readArray;

 writeArray :: STArray s a -> Natural -> a -> ST s ();

 writeArray = Data.Array.ST.writeArray;*}

 code_reserved Haskell Heap

 text {* Monad *}

 code_type Heap (Haskell "Heap.ST/ Heap.RealWorld/ _")

 code_monad bind Haskell

 code_const return (Haskell "return")

 code_const Heap_Monad.raise' (Haskell "error")

 subsubsection {* Scala *}

 code_include Scala "Heap"

 {*import collection.mutable.ArraySeq

 def bind[A, B](f: Unit => A, g: A => Unit => B): Unit => B = (_: Unit) => g (f ()) ()

 class Ref[A](x: A) {

   var value = x

 }

 object Ref {

   def apply[A](x: A): Ref[A] =

     new Ref[A](x)

   def lookup[A](r: Ref[A]): A =

     r.value

   def update[A](r: Ref[A], x: A): Unit =

     { r.value = x }

 }

 object Array {

   def alloc[A](n: Natural.Nat)(x: A): ArraySeq[A] =

     ArraySeq.fill(n.as_Int)(x)

   def make[A](n: Natural.Nat)(f: Natural.Nat => A): ArraySeq[A] =

     ArraySeq.tabulate(n.as_Int)((k: Int) => f(Natural.Nat(k)))

   def len[A](a: ArraySeq[A]): Natural.Nat =

     Natural.Nat(a.length)

   def nth[A](a: ArraySeq[A], n: Natural.Nat): A =

     a(n.as_Int)

   def upd[A](a: ArraySeq[A], n: Natural.Nat, x: A): Unit =

     a.update(n.as_Int, x)

   def freeze[A](a: ArraySeq[A]): List[A] =

     a.toList

 }*}

 code_reserved Scala bind Ref Array

 code_type Heap (Scala "Unit/ =>/ _")

 code_const bind (Scala "Heap.bind")

 code_const return (Scala "('_: Unit)/ =>/ _")

 code_const Heap_Monad.raise' (Scala "!error((_))")

 subsubsection {* Target variants with less units *}

 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_bind = is_const @{const_name bind};

     val is_return = is_const @{const_name return};

     val dummy_name = "";

     val dummy_case_term = IVar NONE;

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

     val (unitt, unitT) = case lookup_const naming @{const_name Unity}

      of SOME unit' => (IConst (unit', (([], []), [])), the (lookup_tyco naming @{type_name 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_bind c

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

           else force t

       | _ => force t;

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

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

     fun imp_monad_bind' (const as (c, (_, tys))) ts = if is_bind 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))

 #> Code_Target.extend_target ("Scala_imp", ("Scala", imp_program))

 end

 *}

 hide_const (open) Heap heap guard raise' fold_map

 end