diff --git a/thys/Van_Emde_Boas_Trees/Imperative_HOL_Time/Heap_Time_Monad.thy b/thys/Van_Emde_Boas_Trees/Imperative_HOL_Time/Heap_Time_Monad.thy
--- a/thys/Van_Emde_Boas_Trees/Imperative_HOL_Time/Heap_Time_Monad.thy
+++ b/thys/Van_Emde_Boas_Trees/Imperative_HOL_Time/Heap_Time_Monad.thy
@@ -1,777 +1,776 @@
 (*  Title:      Imperative_HOL_Time/Heap_Time_Monad.thy
     Author:     Maximilian P. L. Haslbeck & Bohua Zhan, TU Muenchen
 *)
 
 section \<open>A monad with a polymorphic heap and time and primitive reasoning infrastructure\<close>
 
 text \<open>This theory is an adapted version of \<open>Imperative_HOL/Heap_Time_Monad\<close>, where the heap is 
   extended by time bookkeeping.\<close>
 
 theory Heap_Time_Monad
   imports
   "HOL-Imperative_HOL.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 \<times> nat) option"
 
 lemma [code, code del]:
   "(Code_Evaluation.term_of :: 'a::typerep Heap \<Rightarrow> Code_Evaluation.term) = Code_Evaluation.term_of"
   ..
 
 primrec execute :: "'a Heap \<Rightarrow> heap \<Rightarrow> ('a \<times> heap \<times> nat) 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, 1))"
 
 lemma execute_tap [execute_simps]:
   "execute (tap f) h = Some (f h, h, 1)"
   by (simp add: tap_def)
 
 definition heap :: "(heap \<Rightarrow> 'a \<times> heap \<times> nat) \<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 \<times> nat) \<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> vebt_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. AND consume time \<open>n\<close>
 \<close>  
 
 definition effect :: "'a Heap \<Rightarrow> heap \<Rightarrow> heap \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> bool" where
   effect_def: "effect c h h' r n \<longleftrightarrow> execute c h = Some (r, h', n)"
 
 lemma effectI:
   "execute c h = Some (r, h',n) \<Longrightarrow> effect c h h' r n"
   by (simp add: effect_def)
 
 lemma effectE:
   assumes "effect c h h' r n"
   obtains "r = fst (the (execute c h))"
     and "h' = fst (snd (the (execute c h)))"
     and "n = snd (snd (the (execute c h)))"
     and "success c h"
 proof (rule that)
   from assms have *: "execute c h = Some (r, h',n)" by (simp add: effect_def)
   then show "success c h" by (simp add: success_def)
   from * have "fst (the (execute c h)) = r" and "fst (snd (the (execute c h))) = h'"
        and "snd (snd (the (execute c h))) = n"
     by simp_all
   then show "r = fst (the (execute c h))"
     and "h' = fst (snd (the (execute c h)))"
     and "n = snd (snd (the (execute c h)))" by simp_all
 qed
 
 lemma effect_success:
   "effect c h h' r n \<Longrightarrow> success c h"
   by (simp add: effect_def success_def)
 
 lemma success_effectE:
   assumes "success c h"
   obtains r h' n where "effect c h h' r n"
   using assms by (auto simp add: effect_def success_def)
 
 lemma effect_deterministic:
   assumes "effect f h h' a n"
     and "effect f h h'' b n'"
   shows "a = b" and "h' = h''" and "n = n'"
   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 n"
   shows "effect (let x = t in f x) h h' r n"
   using assms by simp
 
 lemma effect_LetE [effect_elims]:
   assumes "effect (let x = t in f x) h h' r n"
   obtains "effect (f t) h h' r n"
   using assms by simp
 
 lemma effect_ifI:
   assumes "c \<Longrightarrow> effect t h h' r n"
     and "\<not> c \<Longrightarrow> effect e h h' r n"
   shows "effect (if c then t else e) h h' r n"
   by (cases c) (simp_all add: assms)
 
 lemma effect_ifE:
   assumes "effect (if c then t else e) h h' r n"
   obtains "c" "effect t h h' r n"
     | "\<not> c" "effect e h h' r n"
   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 1"
   by (rule effectI) (simp add: assms execute_simps)
 
 lemma effect_tapE [effect_elims]:
   assumes "effect (tap f) h h' r n"
   obtains "h' = h" and "r = f h" and "n=1"
   using assms by (rule effectE) (auto simp add: execute_simps)
 
 lemma effect_heapI [effect_intros]:
   assumes  "n = snd (snd (f h))" "h' = fst (snd (f h))" "r = fst (f h)"
   shows "effect (heap f) h h' r n"
   by (rule effectI) (simp add: assms execute_simps)
 
 lemma effect_heapE [effect_elims]:
   assumes "effect (heap f) h h' r n"
   obtains "h' = fst (snd (f h))" and "n = snd (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' = fst (snd (f h))" "n = snd (snd (f h))" "r = fst (f h)"
   shows "effect (guard P f) h h' r n"
   by (rule effectI) (simp add: assms execute_simps)
 
 lemma effect_guardE [effect_elims]:
   assumes "effect (guard P f) h h' r n"
   obtains "h' = fst (snd (f h))" "n = snd (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 (\<lambda>h. (x,h,1))"
 
 lemma execute_return [execute_simps]:
   "execute (return x) = Some \<circ> (\<lambda>h. (x,h,1))"
   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 1"
   by (rule effectI) (simp add: execute_simps)
 
 lemma effect_returnE [effect_elims]:
   assumes "effect (return x) h h' r n"
   obtains "r = x" "h' = h" "n=1"
   using assms by (rule effectE) (simp add: execute_simps)
 
 definition ureturn :: "'a \<Rightarrow> 'a Heap" where
   [code del]: "ureturn x = heap (\<lambda>h. (x,h,0))"
 
 lemma execute_ureturn [execute_simps]:
   "execute (ureturn x) = Some \<circ> (\<lambda>h. (x,h,0))"
   by (simp add: ureturn_def execute_simps)
 
 lemma success_ureturnI [success_intros]:
   "success (ureturn x) h"
   by (rule successI) (simp add: execute_simps)
 
 lemma effect_ureturnI [effect_intros]:
   "h = h' \<Longrightarrow> effect (ureturn x) h h' x 0"
   by (rule effectI) (simp add: execute_simps)
 
 lemma effect_ureturnE [effect_elims]:
   assumes "effect (ureturn x) h h' r n"
   obtains "r = x" "h' = h" "n=0"
   using assms by (rule effectE) (simp add: execute_simps)
 
 definition raise :: "string \<Rightarrow> 'a Heap" where \<comment> \<open>the string is just decoration\<close>
   [code del]: "raise s = Heap (\<lambda>_. None)"
 
 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 n"
   obtains "False"
   using assms by (rule effectE) (simp add: success_def execute_simps)
 
                                   
 fun timeFrame :: "nat \<Rightarrow> ('a \<times> heap \<times> nat) option \<Rightarrow>  ('a \<times> heap \<times> nat) option"  where
   "timeFrame n (Some (r,h,n')) = Some (r, h, n+n')"
 | "timeFrame n None = None"
 
 lemma timeFrame_zero[simp]: "timeFrame 0 h = h" apply(cases h) by auto
 
 
 lemma timeFrame_assoc[simp]: "timeFrame n (timeFrame n' f) = timeFrame (n+n') f"
   by (metis (no_types, lifting) ab_semigroup_add_class.add_ac(1) timeFrame.elims timeFrame.simps(1)) (* refactor proof *)
    
   
   
 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 (r, h', n) \<Rightarrow> timeFrame n (execute (g r) h')
                 | None \<Rightarrow> None)"
   
   
   
 adhoc_overloading
   Monad_Syntax.bind Heap_Time_Monad.bind
 
 lemma execute_bind [execute_simps]:
   "execute f h = Some (x, h',n) \<Longrightarrow> execute (f \<bind> g) h = timeFrame n (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',n) \<Rightarrow> timeFrame n (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
          = timeFrame (snd (snd (the (execute f h)))) (execute (g (fst (the (execute f h)))) (fst (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',n) \<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 n \<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 n" "effect (g r) h' h'' r' n'"
   shows "effect (f \<bind> g) h h'' r' (n+n')"
   using assms
   apply (auto intro!: effectI elim!: effectE successE)
   apply (subst execute_bind, simp_all)
   apply auto
   done
 
 lemma effect_bindE [effect_elims]:
   assumes "effect (f \<bind> g) h h'' r' n"
   obtains h' r n1 n2 where "effect f h h' r n1" "effect (g r) h' h'' r' n2" "n = n1 + n2"
   using assms   apply (auto simp add: effect_def bind_def split: option.split_asm)
   (* To cleanup *)
   by (smt Pair_inject option.distinct(1) option.inject timeFrame.elims)
 
 lemma execute_bind_eq_SomeI:
   assumes "execute f h = Some (x, h',n)"
     and "execute (g x) h' = Some (y, h'',n')"
   shows "execute (f \<bind> g) h = Some (y, h'',n+n')"
   using assms by (simp add: bind_def)
 
 term "return x \<bind> f"    
     
 definition wait :: "nat \<Rightarrow> unit Heap"  where 
    [execute_simps]: "wait n = Heap (\<lambda>h. Some ((),h,n))"   
   
 lemma [simp]: "wait n \<bind> (%_. wait m) = wait (n+m)"  
  unfolding wait_def bind_def by auto 
   
   
 term "wait 1 \<bind> (%_. f x)"   
 term "return x \<bind> f"
   
 lemma ureturn_bind [execute_simps]: "ureturn x \<bind> f = f x"
   apply (rule Heap_eqI) by (simp add: execute_simps)
 
 lemma return_bind [execute_simps]: "return x \<bind> f = (wait 1) \<then> f x"
   apply (rule Heap_eqI) by (simp add: execute_simps)
 
 lemma bind_return [execute_simps]: "f \<bind> return = wait 1 \<then> f"
   by (rule Heap_eqI) (simp add: bind_def execute_simps split: option.splits)
 
 lemma bind_ureturn [execute_simps]: "f \<bind> ureturn = 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 ''assert'')"
 
 lemma execute_assert [execute_simps]:
   "P x \<Longrightarrow> execute (assert P x) h  = Some (x, h, 1)"
   "\<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> n = 1 \<Longrightarrow> effect (assert P x) h h' r n"
   by (rule effectI) (simp add: execute_assert)
  
 lemma effect_assertE [effect_elims]:
   assumes "effect (assert P x) h h' r n"
   obtains "P x" "r = x" "h' = h" "n=1"
   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 execute_simps)
 
 
 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* 'd) option. \<forall>h h' r n. f h = Some (r, h',n) \<longrightarrow> P x h h' r n)"
 proof (rule ccpo.admissibleI)
   fix A :: "('a \<Rightarrow> ('b * 'c * 'd) option) set"
   assume ch: "Complete_Partial_Order.chain option.le_fun A"
     and IH: "\<forall>f\<in>A. \<forall>h h' r n. f h = Some (r, h', n) \<longrightarrow> P x h h' r n"
   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 n. option.lub_fun A h = Some (r, h', n) \<longrightarrow> P x h h' r n"
   proof (intro allI impI)
     fix h h' r n assume "option.lub_fun A h = Some (r, h', n)"
     from flat_lub_in_chain[OF ch' this[unfolded fun_lub_def]]
     have "Some (r, h', n) \<in> {y. \<exists>f\<in>A. y = f h}" by simp
     then have "\<exists>f\<in>A. f h = Some (r, h', n)" by auto
     with IH show "P x h h' r n" by auto
   qed
 qed
 
 lemma admissible_heap: 
   "heap.admissible (\<lambda>f. \<forall>x h h' r n. effect (f x) h h' r n \<longrightarrow> P x h h' r n)"
 proof (rule admissible_fun[OF heap_interpretation])
   fix x
   show "ccpo.admissible Heap_lub Heap_ord (\<lambda>a. \<forall>h h' r n. effect a h h' r n \<longrightarrow> P x h h' r n)"
     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 n. effect a h h' r n \<longrightarrow> P x h h' r n) \<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> nat \<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 n. (\<And>x h h' r n. effect (U f x) h h' r n \<Longrightarrow> P x h h' r n) 
     \<Longrightarrow> effect (U (F f) x) h h' r n \<Longrightarrow> P x h h' r n"
   assumes defined: "effect (U f x) h h' r n"
   shows "P x h h' r n"
   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_time" @{term heap.fixp_fun}
   @{term heap.mono_body} @{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' n where "execute (B g) h = Some (r, h', n)"
         by (metis surjective_pairing)
       then have "execute ?L h = timeFrame n (execute (C r f) h')"
         "execute ?R h = timeFrame n (execute (C r g) h')"
         by (auto simp: execute_bind_case)
       with 2[of r] show ?thesis apply (auto elim: Heap_ordE)
         by (metis Heap_ordE timeFrame.simps(2)) 
     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_Time_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_Time_Monad.raise \<rightharpoonup> (OCaml) "failwith"
 
 
 subsubsection \<open>Haskell\<close>
 
 text \<open>Adaption layer\<close>
 
 code_printing code_module "Heap" \<rightharpoonup> (Haskell)
 \<open>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 Integer a;
 
 newSTRef = Data.STRef.newSTRef;
 readSTRef = Data.STRef.readSTRef;
 writeSTRef = Data.STRef.writeSTRef;
 
 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 = Control.Monad.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_Time_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 make[A](n: BigInt)(f: BigInt => A): T[A] =
-  {
-    val m = n.toInt
-    val a = new T[A](m)
-    for (i <- 0 until m) a(i) = f(i)
+  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 alloc[A](n: BigInt)(x: A): T[A] = make(n)(_ => x)
+  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_Time_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 (op =) @{const_name bind};
     val is_return = curry (op =) @{const_name return};
     val dummy_name = "";
     val dummy_case_term = IVar NONE;
     (*assumption: dummy values are not relevant for serialization*)
     val unitT = @{type_name unit} `%% [];
     val unitt =
       IConst { sym = Code_Symbol.Constant @{const_name Unity}, typargs = [], dicts = [], dom = [],
         annotation = NONE };
     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 }
     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 (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 { 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  TODO *)
 
 (*two additional lemmas, not by Haslbeck and Zhan,
 lemmas just added for this project*)
 
 lemma fold_if_return: "(if b then return c else return d) = return (if b then c else d)" 
   by simp
 
 lemma distrib_if_bind: "do { x \<leftarrow> if b then (c::_ Heap) else d; f x } = (if b then do {x \<leftarrow> c; f x} else do { x\<leftarrow>d; f x })"
   by simp
 
 lemmas heap_monad_laws = bind_return return_bind bind_bind
 end