src/HOL/Imperative_HOL/Heap_Monad.thy
author krauss
Tue Jul 13 12:00:11 2010 +0200 (2010-07-13)
changeset 37792 ba0bc31b90d7
parent 37787 30dc3abf4a58
child 37816 e550439d4422
permissions -rw-r--r--
Heap_Monad uses Monad_Syntax
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(*  Title:      HOL/Imperative_HOL/Heap_Monad.thy
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    Author:     John Matthews, Galois Connections; Alexander Krauss, Lukas Bulwahn & Florian Haftmann, TU Muenchen
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*)
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header {* A monad with a polymorphic heap and primitive reasoning infrastructure *}
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theory Heap_Monad
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imports Heap Monad_Syntax
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begin
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subsection {* The monad *}
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subsubsection {* Monad construction *}
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text {* Monadic heap actions either produce values
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  and transform the heap, or fail *}
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datatype 'a Heap = Heap "heap \<Rightarrow> ('a \<times> heap) option"
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primrec execute :: "'a Heap \<Rightarrow> heap \<Rightarrow> ('a \<times> heap) option" where
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  [code del]: "execute (Heap f) = f"
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lemma Heap_cases [case_names succeed fail]:
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  fixes f and h
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  assumes succeed: "\<And>x h'. execute f h = Some (x, h') \<Longrightarrow> P"
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  assumes fail: "execute f h = None \<Longrightarrow> P"
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  shows P
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  using assms by (cases "execute f h") auto
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lemma Heap_execute [simp]:
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  "Heap (execute f) = f" by (cases f) simp_all
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lemma Heap_eqI:
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  "(\<And>h. execute f h = execute g h) \<Longrightarrow> f = g"
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    by (cases f, cases g) (auto simp: expand_fun_eq)
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ML {* structure Execute_Simps = Named_Thms(
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  val name = "execute_simps"
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  val description = "simplification rules for execute"
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) *}
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setup Execute_Simps.setup
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lemma execute_Let [execute_simps]:
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  "execute (let x = t in f x) = (let x = t in execute (f x))"
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  by (simp add: Let_def)
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subsubsection {* Specialised lifters *}
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definition tap :: "(heap \<Rightarrow> 'a) \<Rightarrow> 'a Heap" where
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  [code del]: "tap f = Heap (\<lambda>h. Some (f h, h))"
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lemma execute_tap [execute_simps]:
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  "execute (tap f) h = Some (f h, h)"
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  by (simp add: tap_def)
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definition heap :: "(heap \<Rightarrow> 'a \<times> heap) \<Rightarrow> 'a Heap" where
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  [code del]: "heap f = Heap (Some \<circ> f)"
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lemma execute_heap [execute_simps]:
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  "execute (heap f) = Some \<circ> f"
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  by (simp add: heap_def)
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definition guard :: "(heap \<Rightarrow> bool) \<Rightarrow> (heap \<Rightarrow> 'a \<times> heap) \<Rightarrow> 'a Heap" where
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  [code del]: "guard P f = Heap (\<lambda>h. if P h then Some (f h) else None)"
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lemma execute_guard [execute_simps]:
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  "\<not> P h \<Longrightarrow> execute (guard P f) h = None"
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  "P h \<Longrightarrow> execute (guard P f) h = Some (f h)"
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  by (simp_all add: guard_def)
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subsubsection {* Predicate classifying successful computations *}
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definition success :: "'a Heap \<Rightarrow> heap \<Rightarrow> bool" where
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  "success f h \<longleftrightarrow> execute f h \<noteq> None"
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lemma successI:
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  "execute f h \<noteq> None \<Longrightarrow> success f h"
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  by (simp add: success_def)
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lemma successE:
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  assumes "success f h"
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  obtains r h' where "r = fst (the (execute c h))"
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    and "h' = snd (the (execute c h))"
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    and "execute f h \<noteq> None"
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  using assms by (simp add: success_def)
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ML {* structure Success_Intros = Named_Thms(
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  val name = "success_intros"
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  val description = "introduction rules for success"
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) *}
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setup Success_Intros.setup
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lemma success_tapI [success_intros]:
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  "success (tap f) h"
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  by (rule successI) (simp add: execute_simps)
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lemma success_heapI [success_intros]:
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  "success (heap f) h"
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  by (rule successI) (simp add: execute_simps)
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lemma success_guardI [success_intros]:
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  "P h \<Longrightarrow> success (guard P f) h"
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  by (rule successI) (simp add: execute_guard)
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lemma success_LetI [success_intros]:
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  "x = t \<Longrightarrow> success (f x) h \<Longrightarrow> success (let x = t in f x) h"
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  by (simp add: Let_def)
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lemma success_ifI:
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  "(c \<Longrightarrow> success t h) \<Longrightarrow> (\<not> c \<Longrightarrow> success e h) \<Longrightarrow>
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    success (if c then t else e) h"
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  by (simp add: success_def)
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subsubsection {* Predicate for a simple relational calculus *}
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text {*
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  The @{text crel} predicate states that when a computation @{text c}
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  runs with the heap @{text h} will result in return value @{text r}
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  and a heap @{text "h'"}, i.e.~no exception occurs.
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*}  
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definition crel :: "'a Heap \<Rightarrow> heap \<Rightarrow> heap \<Rightarrow> 'a \<Rightarrow> bool" where
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  crel_def: "crel c h h' r \<longleftrightarrow> Heap_Monad.execute c h = Some (r, h')"
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lemma crelI:
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  "Heap_Monad.execute c h = Some (r, h') \<Longrightarrow> crel c h h' r"
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  by (simp add: crel_def)
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lemma crelE:
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  assumes "crel c h h' r"
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  obtains "r = fst (the (execute c h))"
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    and "h' = snd (the (execute c h))"
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    and "success c h"
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proof (rule that)
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  from assms have *: "execute c h = Some (r, h')" by (simp add: crel_def)
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  then show "success c h" by (simp add: success_def)
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  from * have "fst (the (execute c h)) = r" and "snd (the (execute c h)) = h'"
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    by simp_all
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  then show "r = fst (the (execute c h))"
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    and "h' = snd (the (execute c h))" by simp_all
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qed
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lemma crel_success:
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  "crel c h h' r \<Longrightarrow> success c h"
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  by (simp add: crel_def success_def)
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lemma success_crelE:
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  assumes "success c h"
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  obtains r h' where "crel c h h' r"
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  using assms by (auto simp add: crel_def success_def)
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lemma crel_deterministic:
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  assumes "crel f h h' a"
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    and "crel f h h'' b"
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  shows "a = b" and "h' = h''"
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  using assms unfolding crel_def by auto
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ML {* structure Crel_Intros = Named_Thms(
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  val name = "crel_intros"
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  val description = "introduction rules for crel"
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) *}
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ML {* structure Crel_Elims = Named_Thms(
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  val name = "crel_elims"
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  val description = "elimination rules for crel"
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) *}
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setup "Crel_Intros.setup #> Crel_Elims.setup"
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lemma crel_LetI [crel_intros]:
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  assumes "x = t" "crel (f x) h h' r"
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  shows "crel (let x = t in f x) h h' r"
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  using assms by simp
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lemma crel_LetE [crel_elims]:
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  assumes "crel (let x = t in f x) h h' r"
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  obtains "crel (f t) h h' r"
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  using assms by simp
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lemma crel_ifI:
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  assumes "c \<Longrightarrow> crel t h h' r"
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    and "\<not> c \<Longrightarrow> crel e h h' r"
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  shows "crel (if c then t else e) h h' r"
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  by (cases c) (simp_all add: assms)
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lemma crel_ifE:
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  assumes "crel (if c then t else e) h h' r"
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  obtains "c" "crel t h h' r"
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    | "\<not> c" "crel e h h' r"
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  using assms by (cases c) simp_all
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lemma crel_tapI [crel_intros]:
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  assumes "h' = h" "r = f h"
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  shows "crel (tap f) h h' r"
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  by (rule crelI) (simp add: assms execute_simps)
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lemma crel_tapE [crel_elims]:
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  assumes "crel (tap f) h h' r"
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  obtains "h' = h" and "r = f h"
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  using assms by (rule crelE) (auto simp add: execute_simps)
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lemma crel_heapI [crel_intros]:
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  assumes "h' = snd (f h)" "r = fst (f h)"
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  shows "crel (heap f) h h' r"
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  by (rule crelI) (simp add: assms execute_simps)
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lemma crel_heapE [crel_elims]:
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  assumes "crel (heap f) h h' r"
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  obtains "h' = snd (f h)" and "r = fst (f h)"
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  using assms by (rule crelE) (simp add: execute_simps)
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lemma crel_guardI [crel_intros]:
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  assumes "P h" "h' = snd (f h)" "r = fst (f h)"
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  shows "crel (guard P f) h h' r"
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  by (rule crelI) (simp add: assms execute_simps)
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lemma crel_guardE [crel_elims]:
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  assumes "crel (guard P f) h h' r"
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  obtains "h' = snd (f h)" "r = fst (f h)" "P h"
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  using assms by (rule crelE)
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    (auto simp add: execute_simps elim!: successE, cases "P h", auto simp add: execute_simps)
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subsubsection {* Monad combinators *}
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definition return :: "'a \<Rightarrow> 'a Heap" where
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  [code del]: "return x = heap (Pair x)"
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lemma execute_return [execute_simps]:
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  "execute (return x) = Some \<circ> Pair x"
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  by (simp add: return_def execute_simps)
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lemma success_returnI [success_intros]:
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  "success (return x) h"
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  by (rule successI) (simp add: execute_simps)
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lemma crel_returnI [crel_intros]:
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  "h = h' \<Longrightarrow> crel (return x) h h' x"
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  by (rule crelI) (simp add: execute_simps)
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lemma crel_returnE [crel_elims]:
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  assumes "crel (return x) h h' r"
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  obtains "r = x" "h' = h"
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  using assms by (rule crelE) (simp add: execute_simps)
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definition raise :: "string \<Rightarrow> 'a Heap" where -- {* the string is just decoration *}
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  [code del]: "raise s = Heap (\<lambda>_. None)"
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lemma execute_raise [execute_simps]:
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  "execute (raise s) = (\<lambda>_. None)"
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  by (simp add: raise_def)
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lemma crel_raiseE [crel_elims]:
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  assumes "crel (raise x) h h' r"
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  obtains "False"
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  using assms by (rule crelE) (simp add: success_def execute_simps)
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definition bind :: "'a Heap \<Rightarrow> ('a \<Rightarrow> 'b Heap) \<Rightarrow> 'b Heap" where
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  [code del]: "bind f g = Heap (\<lambda>h. case execute f h of
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                  Some (x, h') \<Rightarrow> execute (g x) h'
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                | None \<Rightarrow> None)"
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setup {*
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  Adhoc_Overloading.add_variant 
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    @{const_name Monad_Syntax.bindM} @{const_name Heap_Monad.bind}
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*}
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lemma execute_bind [execute_simps]:
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  "execute f h = Some (x, h') \<Longrightarrow> execute (f \<guillemotright>= g) h = execute (g x) h'"
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  "execute f h = None \<Longrightarrow> execute (f \<guillemotright>= g) h = None"
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  by (simp_all add: bind_def)
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lemma execute_bind_success:
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  "success f h \<Longrightarrow> execute (f \<guillemotright>= g) h = execute (g (fst (the (execute f h)))) (snd (the (execute f h)))"
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  by (cases f h rule: Heap_cases) (auto elim!: successE simp add: bind_def)
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lemma success_bind_executeI:
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  "execute f h = Some (x, h') \<Longrightarrow> success (g x) h' \<Longrightarrow> success (f \<guillemotright>= g) h"
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  by (auto intro!: successI elim!: successE simp add: bind_def)
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lemma success_bind_crelI [success_intros]:
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  "crel f h h' x \<Longrightarrow> success (g x) h' \<Longrightarrow> success (f \<guillemotright>= g) h"
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  by (auto simp add: crel_def success_def bind_def)
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lemma crel_bindI [crel_intros]:
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  assumes "crel f h h' r" "crel (g r) h' h'' r'"
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  shows "crel (f \<guillemotright>= g) h h'' r'"
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  using assms
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  apply (auto intro!: crelI elim!: crelE successE)
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  apply (subst execute_bind, simp_all)
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  done
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lemma crel_bindE [crel_elims]:
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  assumes "crel (f \<guillemotright>= g) h h'' r'"
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  obtains h' r where "crel f h h' r" "crel (g r) h' h'' r'"
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  using assms by (auto simp add: crel_def bind_def split: option.split_asm)
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lemma execute_bind_eq_SomeI:
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  assumes "Heap_Monad.execute f h = Some (x, h')"
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    and "Heap_Monad.execute (g x) h' = Some (y, h'')"
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  shows "Heap_Monad.execute (f \<guillemotright>= g) h = Some (y, h'')"
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  using assms by (simp add: bind_def)
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lemma return_bind [simp]: "return x \<guillemotright>= f = f x"
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  by (rule Heap_eqI) (simp add: execute_bind execute_simps)
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lemma bind_return [simp]: "f \<guillemotright>= return = f"
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  by (rule Heap_eqI) (simp add: bind_def execute_simps split: option.splits)
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lemma bind_bind [simp]: "(f \<guillemotright>= g) \<guillemotright>= k = f \<guillemotright>= (\<lambda>x. g x \<guillemotright>= k)"
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  by (rule Heap_eqI) (simp add: bind_def execute_simps split: option.splits)
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lemma raise_bind [simp]: "raise e \<guillemotright>= f = raise e"
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  by (rule Heap_eqI) (simp add: execute_simps)
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subsection {* Generic combinators *}
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subsubsection {* Assertions *}
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definition assert :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a Heap" where
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  "assert P x = (if P x then return x else raise ''assert'')"
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lemma execute_assert [execute_simps]:
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  "P x \<Longrightarrow> execute (assert P x) h = Some (x, h)"
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  "\<not> P x \<Longrightarrow> execute (assert P x) h = None"
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  by (simp_all add: assert_def execute_simps)
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lemma success_assertI [success_intros]:
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  "P x \<Longrightarrow> success (assert P x) h"
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  by (rule successI) (simp add: execute_assert)
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lemma crel_assertI [crel_intros]:
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  "P x \<Longrightarrow> h' = h \<Longrightarrow> r = x \<Longrightarrow> crel (assert P x) h h' r"
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  by (rule crelI) (simp add: execute_assert)
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lemma crel_assertE [crel_elims]:
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  assumes "crel (assert P x) h h' r"
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  obtains "P x" "r = x" "h' = h"
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  using assms by (rule crelE) (cases "P x", simp_all add: execute_assert success_def)
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lemma assert_cong [fundef_cong]:
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  assumes "P = P'"
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  assumes "\<And>x. P' x \<Longrightarrow> f x = f' x"
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  shows "(assert P x >>= f) = (assert P' x >>= f')"
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  by (rule Heap_eqI) (insert assms, simp add: assert_def)
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subsubsection {* Plain lifting *}
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definition lift :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b Heap" where
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  "lift f = return o f"
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lemma lift_collapse [simp]:
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  "lift f x = return (f x)"
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  by (simp add: lift_def)
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lemma bind_lift:
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  "(f \<guillemotright>= lift g) = (f \<guillemotright>= (\<lambda>x. return (g x)))"
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  by (simp add: lift_def comp_def)
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subsubsection {* Iteration -- warning: this is rarely useful! *}
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primrec fold_map :: "('a \<Rightarrow> 'b Heap) \<Rightarrow> 'a list \<Rightarrow> 'b list Heap" where
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  "fold_map f [] = return []"
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| "fold_map f (x # xs) = do {
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     y \<leftarrow> f x;
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     ys \<leftarrow> fold_map f xs;
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     return (y # ys)
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   }"
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   378
lemma fold_map_append:
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  "fold_map f (xs @ ys) = fold_map f xs \<guillemotright>= (\<lambda>xs. fold_map f ys \<guillemotright>= (\<lambda>ys. return (xs @ ys)))"
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  by (induct xs) simp_all
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   381
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   382
lemma execute_fold_map_unchanged_heap [execute_simps]:
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  assumes "\<And>x. x \<in> set xs \<Longrightarrow> \<exists>y. execute (f x) h = Some (y, h)"
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  shows "execute (fold_map f xs) h =
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   385
    Some (List.map (\<lambda>x. fst (the (execute (f x) h))) xs, h)"
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using assms proof (induct xs)
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  case Nil show ?case by (simp add: execute_simps)
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next
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  case (Cons x xs)
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   390
  from Cons.prems obtain y
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    where y: "execute (f x) h = Some (y, h)" by auto
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  moreover from Cons.prems Cons.hyps have "execute (fold_map f xs) h =
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    Some (map (\<lambda>x. fst (the (execute (f x) h))) xs, h)" by auto
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  ultimately show ?case by (simp, simp only: execute_bind(1), simp add: execute_simps)
haftmann@37754
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qed
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subsection {* Code generator setup *}
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   398
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   399
subsubsection {* Logical intermediate layer *}
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   400
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primrec raise' :: "String.literal \<Rightarrow> 'a Heap" where
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  [code del, code_post]: "raise' (STR s) = raise s"
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   403
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   404
lemma raise_raise' [code_inline]:
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  "raise s = raise' (STR s)"
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   406
  by simp
haftmann@26182
   407
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   408
code_datatype raise' -- {* avoid @{const "Heap"} formally *}
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   409
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   410
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   411
subsubsection {* SML and OCaml *}
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   412
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   413
code_type Heap (SML "unit/ ->/ _")
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code_const "op \<guillemotright>=" (SML "!(fn/ f'_/ =>/ fn/ ()/ =>/ f'_/ (_/ ())/ ())")
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code_const return (SML "!(fn/ ()/ =>/ _)")
haftmann@37709
   416
code_const Heap_Monad.raise' (SML "!(raise/ Fail/ _)")
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   417
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   418
code_type Heap (OCaml "unit/ ->/ _")
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   419
code_const "op \<guillemotright>=" (OCaml "!(fun/ f'_/ ()/ ->/ f'_/ (_/ ())/ ())")
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   420
code_const return (OCaml "!(fun/ ()/ ->/ _)")
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   421
code_const Heap_Monad.raise' (OCaml "failwith/ _")
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   422
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   423
setup {*
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   424
haftmann@31871
   425
let
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   426
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   427
open Code_Thingol;
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   428
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   429
fun imp_program naming =
haftmann@27707
   430
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   431
  let
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   432
    fun is_const c = case lookup_const naming c
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   433
     of SOME c' => (fn c'' => c' = c'')
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   434
      | NONE => K false;
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   435
    val is_bind = is_const @{const_name bind};
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   436
    val is_return = is_const @{const_name return};
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   437
    val dummy_name = "";
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   438
    val dummy_type = ITyVar dummy_name;
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   439
    val dummy_case_term = IVar NONE;
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   440
    (*assumption: dummy values are not relevant for serialization*)
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   441
    val unitt = case lookup_const naming @{const_name Unity}
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   442
     of SOME unit' => IConst (unit', (([], []), []))
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   443
      | NONE => error ("Must include " ^ @{const_name Unity} ^ " in generated constants.");
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   444
    fun dest_abs ((v, ty) `|=> t, _) = ((v, ty), t)
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   445
      | dest_abs (t, ty) =
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   446
          let
haftmann@31871
   447
            val vs = fold_varnames cons t [];
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   448
            val v = Name.variant vs "x";
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   449
            val ty' = (hd o fst o unfold_fun) ty;
haftmann@31893
   450
          in ((SOME v, ty'), t `$ IVar (SOME v)) end;
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   451
    fun force (t as IConst (c, _) `$ t') = if is_return c
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   452
          then t' else t `$ unitt
haftmann@31871
   453
      | force t = t `$ unitt;
haftmann@31871
   454
    fun tr_bind' [(t1, _), (t2, ty2)] =
haftmann@31871
   455
      let
haftmann@31871
   456
        val ((v, ty), t) = dest_abs (t2, ty2);
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   457
      in ICase (((force t1, ty), [(IVar v, tr_bind'' t)]), dummy_case_term) end
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   458
    and tr_bind'' t = case unfold_app t
haftmann@37754
   459
         of (IConst (c, (_, ty1 :: ty2 :: _)), [x1, x2]) => if is_bind c
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   460
              then tr_bind' [(x1, ty1), (x2, ty2)]
haftmann@31871
   461
              else force t
haftmann@31871
   462
          | _ => force t;
haftmann@31893
   463
    fun imp_monad_bind'' ts = (SOME dummy_name, dummy_type) `|=> ICase (((IVar (SOME dummy_name), dummy_type),
haftmann@31871
   464
      [(unitt, tr_bind' ts)]), dummy_case_term)
haftmann@37754
   465
    and imp_monad_bind' (const as (c, (_, tys))) ts = if is_bind c then case (ts, tys)
haftmann@31871
   466
       of ([t1, t2], ty1 :: ty2 :: _) => imp_monad_bind'' [(t1, ty1), (t2, ty2)]
haftmann@31871
   467
        | ([t1, t2, t3], ty1 :: ty2 :: _) => imp_monad_bind'' [(t1, ty1), (t2, ty2)] `$ t3
haftmann@31871
   468
        | (ts, _) => imp_monad_bind (eta_expand 2 (const, ts))
haftmann@31871
   469
      else IConst const `$$ map imp_monad_bind ts
haftmann@31871
   470
    and imp_monad_bind (IConst const) = imp_monad_bind' const []
haftmann@31871
   471
      | imp_monad_bind (t as IVar _) = t
haftmann@31871
   472
      | imp_monad_bind (t as _ `$ _) = (case unfold_app t
haftmann@31871
   473
         of (IConst const, ts) => imp_monad_bind' const ts
haftmann@31871
   474
          | (t, ts) => imp_monad_bind t `$$ map imp_monad_bind ts)
haftmann@31871
   475
      | imp_monad_bind (v_ty `|=> t) = v_ty `|=> imp_monad_bind t
haftmann@31871
   476
      | imp_monad_bind (ICase (((t, ty), pats), t0)) = ICase
haftmann@31871
   477
          (((imp_monad_bind t, ty),
haftmann@31871
   478
            (map o pairself) imp_monad_bind pats),
haftmann@31871
   479
              imp_monad_bind t0);
haftmann@28663
   480
haftmann@31871
   481
  in (Graph.map_nodes o map_terms_stmt) imp_monad_bind end;
haftmann@27707
   482
haftmann@27707
   483
in
haftmann@27707
   484
haftmann@31871
   485
Code_Target.extend_target ("SML_imp", ("SML", imp_program))
haftmann@31871
   486
#> Code_Target.extend_target ("OCaml_imp", ("OCaml", imp_program))
haftmann@27707
   487
haftmann@27707
   488
end
haftmann@31871
   489
haftmann@27707
   490
*}
haftmann@27707
   491
haftmann@26182
   492
haftmann@26182
   493
subsubsection {* Haskell *}
haftmann@26182
   494
haftmann@26182
   495
text {* Adaption layer *}
haftmann@26182
   496
haftmann@29793
   497
code_include Haskell "Heap"
haftmann@26182
   498
{*import qualified Control.Monad;
haftmann@26182
   499
import qualified Control.Monad.ST;
haftmann@26182
   500
import qualified Data.STRef;
haftmann@26182
   501
import qualified Data.Array.ST;
haftmann@26182
   502
haftmann@27695
   503
type RealWorld = Control.Monad.ST.RealWorld;
haftmann@26182
   504
type ST s a = Control.Monad.ST.ST s a;
haftmann@26182
   505
type STRef s a = Data.STRef.STRef s a;
haftmann@27673
   506
type STArray s a = Data.Array.ST.STArray s Int a;
haftmann@26182
   507
haftmann@26182
   508
newSTRef = Data.STRef.newSTRef;
haftmann@26182
   509
readSTRef = Data.STRef.readSTRef;
haftmann@26182
   510
writeSTRef = Data.STRef.writeSTRef;
haftmann@26182
   511
haftmann@27673
   512
newArray :: (Int, Int) -> a -> ST s (STArray s a);
haftmann@26182
   513
newArray = Data.Array.ST.newArray;
haftmann@26182
   514
haftmann@27673
   515
newListArray :: (Int, Int) -> [a] -> ST s (STArray s a);
haftmann@26182
   516
newListArray = Data.Array.ST.newListArray;
haftmann@26182
   517
haftmann@27673
   518
lengthArray :: STArray s a -> ST s Int;
haftmann@27673
   519
lengthArray a = Control.Monad.liftM snd (Data.Array.ST.getBounds a);
haftmann@26182
   520
haftmann@27673
   521
readArray :: STArray s a -> Int -> ST s a;
haftmann@26182
   522
readArray = Data.Array.ST.readArray;
haftmann@26182
   523
haftmann@27673
   524
writeArray :: STArray s a -> Int -> a -> ST s ();
haftmann@26182
   525
writeArray = Data.Array.ST.writeArray;*}
haftmann@26182
   526
haftmann@29793
   527
code_reserved Haskell Heap
haftmann@26182
   528
haftmann@26182
   529
text {* Monad *}
haftmann@26182
   530
haftmann@29793
   531
code_type Heap (Haskell "Heap.ST/ Heap.RealWorld/ _")
krauss@37792
   532
code_monad bind Haskell
haftmann@26182
   533
code_const return (Haskell "return")
haftmann@37709
   534
code_const Heap_Monad.raise' (Haskell "error/ _")
haftmann@26182
   535
haftmann@37758
   536
hide_const (open) Heap heap guard raise' fold_map
haftmann@37724
   537
haftmann@26170
   538
end