(* Title: HOL/Imperative_HOL/Heap_Monad.thy
Author: John Matthews, Galois Connections; Alexander Krauss, Lukas Bulwahn & Florian Haftmann, TU Muenchen
*)
section \<open>A monad with a polymorphic heap and primitive reasoning infrastructure\<close>
theory Heap_Monad
imports
Heap
"HOL-Library.Monad_Syntax"
begin
subsection \<open>The monad\<close>
subsubsection \<open>Monad construction\<close>
text \<open>Monadic heap actions either produce values
and transform the heap, or fail\<close>
datatype 'a Heap = Heap "heap \<Rightarrow> ('a \<times> heap) option"
declare [[code drop: "Code_Evaluation.term_of :: 'a::typerep Heap \<Rightarrow> Code_Evaluation.term"]]
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: fun_eq_iff)
named_theorems execute_simps "simplification rules for execute"
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 \<open>Specialised lifters\<close>
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 \<open>Predicate classifying successful computations\<close>
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 "execute f h = Some (r, h')"
using assms by (auto simp: success_def)
named_theorems success_intros "introduction rules for success"
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 \<open>Predicate for a simple relational calculus\<close>
text \<open>
The \<open>effect\<close> predicate states that when a computation \<open>c\<close>
runs with the heap \<open>h\<close> will result in return value \<open>r\<close>
and a heap \<open>h'\<close>, i.e.~no exception occurs.
\<close>
definition effect :: "'a Heap \<Rightarrow> heap \<Rightarrow> heap \<Rightarrow> 'a \<Rightarrow> bool" where
effect_def: "effect c h h' r \<longleftrightarrow> execute c h = Some (r, h')"
lemma effectI:
"execute c h = Some (r, h') \<Longrightarrow> effect c h h' r"
by (simp add: effect_def)
lemma effectE:
assumes "effect 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: effect_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 effect_success:
"effect c h h' r \<Longrightarrow> success c h"
by (simp add: effect_def success_def)
lemma success_effectE:
assumes "success c h"
obtains r h' where "effect c h h' r"
using assms by (auto simp add: effect_def success_def)
lemma effect_deterministic:
assumes "effect f h h' a"
and "effect f h h'' b"
shows "a = b" and "h' = h''"
using assms unfolding effect_def by auto
named_theorems effect_intros "introduction rules for effect"
and effect_elims "elimination rules for effect"
lemma effect_LetI [effect_intros]:
assumes "x = t" "effect (f x) h h' r"
shows "effect (let x = t in f x) h h' r"
using assms by simp
lemma effect_LetE [effect_elims]:
assumes "effect (let x = t in f x) h h' r"
obtains "effect (f t) h h' r"
using assms by simp
lemma effect_ifI:
assumes "c \<Longrightarrow> effect t h h' r"
and "\<not> c \<Longrightarrow> effect e h h' r"
shows "effect (if c then t else e) h h' r"
by (cases c) (simp_all add: assms)
lemma effect_ifE:
assumes "effect (if c then t else e) h h' r"
obtains "c" "effect t h h' r"
| "\<not> c" "effect e h h' r"
using assms by (cases c) simp_all
lemma effect_tapI [effect_intros]:
assumes "h' = h" "r = f h"
shows "effect (tap f) h h' r"
by (rule effectI) (simp add: assms execute_simps)
lemma effect_tapE [effect_elims]:
assumes "effect (tap f) h h' r"
obtains "h' = h" and "r = f h"
using assms by (rule effectE) (auto simp add: execute_simps)
lemma effect_heapI [effect_intros]:
assumes "h' = snd (f h)" "r = fst (f h)"
shows "effect (heap f) h h' r"
by (rule effectI) (simp add: assms execute_simps)
lemma effect_heapE [effect_elims]:
assumes "effect (heap f) h h' r"
obtains "h' = snd (f h)" and "r = fst (f h)"
using assms by (rule effectE) (simp add: execute_simps)
lemma effect_guardI [effect_intros]:
assumes "P h" "h' = snd (f h)" "r = fst (f h)"
shows "effect (guard P f) h h' r"
by (rule effectI) (simp add: assms execute_simps)
lemma effect_guardE [effect_elims]:
assumes "effect (guard P f) h h' r"
obtains "h' = snd (f h)" "r = fst (f h)" "P h"
using assms by (rule effectE)
(auto simp add: execute_simps elim!: successE, cases "P h", auto simp add: execute_simps)
subsubsection \<open>Monad combinators\<close>
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 effect_returnI [effect_intros]:
"h = h' \<Longrightarrow> effect (return x) h h' x"
by (rule effectI) (simp add: execute_simps)
lemma effect_returnE [effect_elims]:
assumes "effect (return x) h h' r"
obtains "r = x" "h' = h"
using assms by (rule effectE) (simp add: execute_simps)
definition raise :: "String.literal \<Rightarrow> 'a Heap" \<comment> \<open>the literal is just decoration\<close>
where "raise s = Heap (\<lambda>_. None)"
code_datatype raise \<comment> \<open>avoid \<^const>\<open>Heap\<close> formally\<close>
lemma execute_raise [execute_simps]:
"execute (raise s) = (\<lambda>_. None)"
by (simp add: raise_def)
lemma effect_raiseE [effect_elims]:
assumes "effect (raise x) h h' r"
obtains "False"
using assms by (rule effectE) (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)"
adhoc_overloading
Monad_Syntax.bind Heap_Monad.bind
lemma execute_bind [execute_simps]:
"execute f h = Some (x, h') \<Longrightarrow> execute (f \<bind> g) h = execute (g x) h'"
"execute f h = None \<Longrightarrow> execute (f \<bind> g) h = None"
by (simp_all add: bind_def)
lemma execute_bind_case:
"execute (f \<bind> 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 \<bind> 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 \<bind> g) h"
by (auto intro!: successI elim: successE simp add: bind_def)
lemma success_bind_effectI [success_intros]:
"effect f h h' x \<Longrightarrow> success (g x) h' \<Longrightarrow> success (f \<bind> g) h"
by (auto simp add: effect_def success_def bind_def)
lemma effect_bindI [effect_intros]:
assumes "effect f h h' r" "effect (g r) h' h'' r'"
shows "effect (f \<bind> g) h h'' r'"
using assms
apply (auto intro!: effectI elim!: effectE successE)
apply (subst execute_bind, simp_all)
done
lemma effect_bindE [effect_elims]:
assumes "effect (f \<bind> g) h h'' r'"
obtains h' r where "effect f h h' r" "effect (g r) h' h'' r'"
using assms by (auto simp add: effect_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 \<bind> g) h = Some (y, h'')"
using assms by (simp add: bind_def)
lemma return_bind [simp]: "return x \<bind> f = f x"
by (rule Heap_eqI) (simp add: execute_simps)
lemma bind_return [simp]: "f \<bind> return = f"
by (rule Heap_eqI) (simp add: bind_def execute_simps split: option.splits)
lemma bind_bind [simp]: "(f \<bind> g) \<bind> k = (f :: 'a Heap) \<bind> (\<lambda>x. g x \<bind> k)"
by (rule Heap_eqI) (simp add: bind_def execute_simps split: option.splits)
lemma raise_bind [simp]: "raise e \<bind> f = raise e"
by (rule Heap_eqI) (simp add: execute_simps)
subsection \<open>Generic combinators\<close>
subsubsection \<open>Assertions\<close>
definition assert :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a Heap" where
"assert P x = (if P x then return x else raise STR ''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 effect_assertI [effect_intros]:
"P x \<Longrightarrow> h' = h \<Longrightarrow> r = x \<Longrightarrow> effect (assert P x) h h' r"
by (rule effectI) (simp add: execute_assert)
lemma effect_assertE [effect_elims]:
assumes "effect (assert P x) h h' r"
obtains "P x" "r = x" "h' = h"
using assms by (rule effectE) (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 \<bind> f) = (assert P' x \<bind> f')"
by (rule Heap_eqI) (insert assms, simp add: assert_def)
subsubsection \<open>Plain lifting\<close>
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 \<bind> lift g) = (f \<bind> (\<lambda>x. return (g x)))"
by (simp add: lift_def comp_def)
subsubsection \<open>Iteration -- warning: this is rarely useful!\<close>
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 \<bind> (\<lambda>xs. fold_map f ys \<bind> (\<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 \<open>Partial function definition setup\<close>
definition Heap_ord :: "'a Heap \<Rightarrow> 'a Heap \<Rightarrow> bool" where
"Heap_ord = img_ord execute (fun_ord option_ord)"
definition Heap_lub :: "'a Heap set \<Rightarrow> 'a Heap" where
"Heap_lub = img_lub execute Heap (fun_lub (flat_lub None))"
lemma Heap_lub_empty: "Heap_lub {} = Heap Map.empty"
by(simp add: Heap_lub_def img_lub_def fun_lub_def flat_lub_def)
lemma heap_interpretation: "partial_function_definitions Heap_ord Heap_lub"
proof -
have "partial_function_definitions (fun_ord option_ord) (fun_lub (flat_lub None))"
by (rule partial_function_lift) (rule flat_interpretation)
then have "partial_function_definitions (img_ord execute (fun_ord option_ord))
(img_lub execute Heap (fun_lub (flat_lub None)))"
by (rule partial_function_image) (auto intro: Heap_eqI)
then show "partial_function_definitions Heap_ord Heap_lub"
by (simp only: Heap_ord_def Heap_lub_def)
qed
interpretation heap: partial_function_definitions Heap_ord Heap_lub
rewrites "Heap_lub {} \<equiv> Heap Map.empty"
by (fact heap_interpretation)(simp add: Heap_lub_empty)
lemma heap_step_admissible:
"option.admissible
(\<lambda>f:: 'a => ('b * 'c) option. \<forall>h h' r. f h = Some (r, h') \<longrightarrow> P x h h' r)"
proof (rule ccpo.admissibleI)
fix A :: "('a \<Rightarrow> ('b * 'c) option) set"
assume ch: "Complete_Partial_Order.chain option.le_fun A"
and IH: "\<forall>f\<in>A. \<forall>h h' r. f h = Some (r, h') \<longrightarrow> P x h h' r"
from ch have ch': "\<And>x. Complete_Partial_Order.chain option_ord {y. \<exists>f\<in>A. y = f x}" by (rule chain_fun)
show "\<forall>h h' r. option.lub_fun A h = Some (r, h') \<longrightarrow> P x h h' r"
proof (intro allI impI)
fix h h' r assume "option.lub_fun A h = Some (r, h')"
from flat_lub_in_chain[OF ch' this[unfolded fun_lub_def]]
have "Some (r, h') \<in> {y. \<exists>f\<in>A. y = f h}" by simp
then have "\<exists>f\<in>A. f h = Some (r, h')" by auto
with IH show "P x h h' r" by auto
qed
qed
lemma admissible_heap:
"heap.admissible (\<lambda>f. \<forall>x h h' r. effect (f x) h h' r \<longrightarrow> P x h h' r)"
proof (rule admissible_fun[OF heap_interpretation])
fix x
show "ccpo.admissible Heap_lub Heap_ord (\<lambda>a. \<forall>h h' r. effect a h h' r \<longrightarrow> P x h h' r)"
unfolding Heap_ord_def Heap_lub_def
proof (intro admissible_image partial_function_lift flat_interpretation)
show "option.admissible ((\<lambda>a. \<forall>h h' r. effect a h h' r \<longrightarrow> P x h h' r) \<circ> Heap)"
unfolding comp_def effect_def execute.simps
by (rule heap_step_admissible)
qed (auto simp add: Heap_eqI)
qed
lemma fixp_induct_heap:
fixes F :: "'c \<Rightarrow> 'c" and
U :: "'c \<Rightarrow> 'b \<Rightarrow> 'a Heap" and
C :: "('b \<Rightarrow> 'a Heap) \<Rightarrow> 'c" and
P :: "'b \<Rightarrow> heap \<Rightarrow> heap \<Rightarrow> 'a \<Rightarrow> bool"
assumes mono: "\<And>x. monotone (fun_ord Heap_ord) Heap_ord (\<lambda>f. U (F (C f)) x)"
assumes eq: "f \<equiv> C (ccpo.fixp (fun_lub Heap_lub) (fun_ord Heap_ord) (\<lambda>f. U (F (C f))))"
assumes inverse2: "\<And>f. U (C f) = f"
assumes step: "\<And>f x h h' r. (\<And>x h h' r. effect (U f x) h h' r \<Longrightarrow> P x h h' r)
\<Longrightarrow> effect (U (F f) x) h h' r \<Longrightarrow> P x h h' r"
assumes defined: "effect (U f x) h h' r"
shows "P x h h' r"
using step defined heap.fixp_induct_uc[of U F C, OF mono eq inverse2 admissible_heap, of P]
unfolding effect_def execute.simps
by blast
declaration \<open>Partial_Function.init "heap" \<^term>\<open>heap.fixp_fun\<close>
\<^term>\<open>heap.mono_body\<close> @{thm heap.fixp_rule_uc} @{thm heap.fixp_induct_uc}
(SOME @{thm fixp_induct_heap})\<close>
abbreviation "mono_Heap \<equiv> monotone (fun_ord Heap_ord) Heap_ord"
lemma Heap_ordI:
assumes "\<And>h. execute x h = None \<or> execute x h = execute y h"
shows "Heap_ord x y"
using assms unfolding Heap_ord_def img_ord_def fun_ord_def flat_ord_def
by blast
lemma Heap_ordE:
assumes "Heap_ord x y"
obtains "execute x h = None" | "execute x h = execute y h"
using assms unfolding Heap_ord_def img_ord_def fun_ord_def flat_ord_def
by atomize_elim blast
lemma bind_mono [partial_function_mono]:
assumes mf: "mono_Heap B" and mg: "\<And>y. mono_Heap (\<lambda>f. C y f)"
shows "mono_Heap (\<lambda>f. B f \<bind> (\<lambda>y. C y f))"
proof (rule monotoneI)
fix f g :: "'a \<Rightarrow> 'b Heap" assume fg: "fun_ord Heap_ord f g"
from mf
have 1: "Heap_ord (B f) (B g)" by (rule monotoneD) (rule fg)
from mg
have 2: "\<And>y'. Heap_ord (C y' f) (C y' g)" by (rule monotoneD) (rule fg)
have "Heap_ord (B f \<bind> (\<lambda>y. C y f)) (B g \<bind> (\<lambda>y. C y f))"
(is "Heap_ord ?L ?R")
proof (rule Heap_ordI)
fix h
from 1 show "execute ?L h = None \<or> execute ?L h = execute ?R h"
by (rule Heap_ordE[where h = h]) (auto simp: execute_bind_case)
qed
also
have "Heap_ord (B g \<bind> (\<lambda>y'. C y' f)) (B g \<bind> (\<lambda>y'. C y' g))"
(is "Heap_ord ?L ?R")
proof (rule Heap_ordI)
fix h
show "execute ?L h = None \<or> execute ?L h = execute ?R h"
proof (cases "execute (B g) h")
case None
then have "execute ?L h = None" by (auto simp: execute_bind_case)
thus ?thesis ..
next
case Some
then obtain r h' where "execute (B g) h = Some (r, h')"
by (metis surjective_pairing)
then have "execute ?L h = execute (C r f) h'"
"execute ?R h = execute (C r g) h'"
by (auto simp: execute_bind_case)
with 2[of r] show ?thesis by (auto elim: Heap_ordE)
qed
qed
finally (heap.leq_trans)
show "Heap_ord (B f \<bind> (\<lambda>y. C y f)) (B g \<bind> (\<lambda>y'. C y' g))" .
qed
subsection \<open>Code generator setup\<close>
subsubsection \<open>SML and OCaml\<close>
code_printing type_constructor Heap \<rightharpoonup> (SML) "(unit/ ->/ _)"
code_printing constant bind \<rightharpoonup> (SML) "!(fn/ f'_/ =>/ fn/ ()/ =>/ f'_/ (_/ ())/ ())"
code_printing constant return \<rightharpoonup> (SML) "!(fn/ ()/ =>/ _)"
code_printing constant Heap_Monad.raise \<rightharpoonup> (SML) "!(raise/ Fail/ _)"
code_printing type_constructor Heap \<rightharpoonup> (OCaml) "(unit/ ->/ _)"
code_printing constant bind \<rightharpoonup> (OCaml) "!(fun/ f'_/ ()/ ->/ f'_/ (_/ ())/ ())"
code_printing constant return \<rightharpoonup> (OCaml) "!(fun/ ()/ ->/ _)"
code_printing constant Heap_Monad.raise \<rightharpoonup> (OCaml) "failwith"
subsubsection \<open>Haskell\<close>
text \<open>Adaption layer\<close>
code_printing code_module "Heap" \<rightharpoonup> (Haskell)
\<open>
module Heap(ST, RealWorld, STRef, newSTRef, readSTRef, writeSTRef,
STArray, newArray, newListArray, newFunArray, lengthArray, readArray, writeArray) where
import Control.Monad(liftM)
import Control.Monad.ST(RealWorld, ST)
import Data.STRef(STRef, newSTRef, readSTRef, writeSTRef)
import qualified Data.Array.ST
type STArray s a = Data.Array.ST.STArray s Integer a
newArray :: Integer -> a -> ST s (STArray s a)
newArray k = Data.Array.ST.newArray (0, k - 1)
newListArray :: [a] -> ST s (STArray s a)
newListArray xs = Data.Array.ST.newListArray (0, (fromInteger . toInteger . length) xs - 1) xs
newFunArray :: Integer -> (Integer -> a) -> ST s (STArray s a)
newFunArray k f = Data.Array.ST.newListArray (0, k - 1) (map f [0..k-1])
lengthArray :: STArray s a -> ST s Integer
lengthArray a = liftM (\(_, l) -> l + 1) (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\<close>
code_reserved (Haskell) Heap
text \<open>Monad\<close>
code_printing type_constructor Heap \<rightharpoonup> (Haskell) "Heap.ST/ Heap.RealWorld/ _"
code_monad bind Haskell
code_printing constant return \<rightharpoonup> (Haskell) "return"
code_printing constant Heap_Monad.raise \<rightharpoonup> (Haskell) "error"
subsubsection \<open>Scala\<close>
code_printing code_module "Heap" \<rightharpoonup> (Scala)
\<open>object Heap {
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 {
class T[A](n: Int)
{
val array: Array[AnyRef] = new Array[AnyRef](n)
def apply(i: Int): A = array(i).asInstanceOf[A]
def update(i: Int, x: A): Unit = array(i) = x.asInstanceOf[AnyRef]
def length: Int = array.length
def toList: List[A] = array.toList.asInstanceOf[List[A]]
override def toString: String = array.mkString("Array.T(", ",", ")")
}
def init[A](n: Int)(f: Int => A): T[A] = {
val a = new T[A](n)
for (i <- 0 until n) a(i) = f(i)
a
}
def make[A](n: BigInt)(f: BigInt => A): T[A] = init(n.toInt)((i: Int) => f(BigInt(i)))
def alloc[A](n: BigInt)(x: A): T[A] = init(n.toInt)(_ => x)
def len[A](a: T[A]): BigInt = BigInt(a.length)
def nth[A](a: T[A], n: BigInt): A = a(n.toInt)
def upd[A](a: T[A], n: BigInt, x: A): Unit = a.update(n.toInt, x)
def freeze[A](a: T[A]): List[A] = a.toList
}
\<close>
code_reserved (Scala) Heap Ref Array
code_printing type_constructor Heap \<rightharpoonup> (Scala) "(Unit/ =>/ _)"
code_printing constant bind \<rightharpoonup> (Scala) "Heap.bind"
code_printing constant return \<rightharpoonup> (Scala) "('_: Unit)/ =>/ _"
code_printing constant Heap_Monad.raise \<rightharpoonup> (Scala) "!sys.error((_))"
subsubsection \<open>Target variants with less units\<close>
setup \<open>
let
open Code_Thingol;
val imp_program =
let
val is_bind = curry (=) \<^const_name>\<open>bind\<close>;
val is_return = curry (=) \<^const_name>\<open>return\<close>;
val dummy_name = "";
val dummy_case_term = IVar NONE;
(*assumption: dummy values are not relevant for serialization*)
val unitT = \<^type_name>\<open>unit\<close> `%% [];
val unitt =
IConst { sym = Code_Symbol.Constant \<^const_name>\<open>Unity\<close>, typargs = [], dicts = [], dom = [],
annotation = NONE, range = unitT };
fun dest_abs ((v, ty) `|=> (t, _), _) = ((v, ty), t)
| dest_abs (t, ty) =
let
val vs = fold_varnames cons t [];
val v = singleton (Name.variant_list 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 { sym = Code_Symbol.Constant 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 { term = force t1, typ = ty, clauses = [(IVar v, tr_bind' t)], primitive = dummy_case_term } end
and tr_bind' t = case unfold_app t
of (IConst { sym = Code_Symbol.Constant c, dom = 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 { term = IVar (SOME dummy_name), typ = unitT, clauses = [(unitt, tr_bind'' ts)], primitive = dummy_case_term }, unitT)
fun imp_monad_bind' (const as { sym = Code_Symbol.Constant c, dom = dom, ... }) ts = if is_bind c then case (ts, dom)
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 (saturated_application 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 `|=> apfst imp_monad_bind t
| imp_monad_bind (ICase { term = t, typ = ty, clauses = clauses, primitive = t0 }) =
ICase { term = imp_monad_bind t, typ = ty,
clauses = (map o apply2) imp_monad_bind clauses, primitive = imp_monad_bind t0 };
in (Code_Symbol.Graph.map o K o map_terms_stmt) imp_monad_bind end;
in
Code_Target.add_derived_target ("SML_imp", [("SML", imp_program)])
#> Code_Target.add_derived_target ("OCaml_imp", [("OCaml", imp_program)])
#> Code_Target.add_derived_target ("Scala_imp", [("Scala", imp_program)])
end
\<close>
hide_const (open) Heap heap guard fold_map
end