theory Relational 

 imports Array Ref

 begin

 section{* Definition of the Relational framework *}

 text {* The crel predicate states that when a computation c runs with the heap h

   will result in return value r and a heap h' (if no exception occurs). *}  

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

 where

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

 lemma crel_def: -- FIXME

   "crel c h h' r \<longleftrightarrow> (Inl r, h') = Heap_Monad.execute c h"

   unfolding crel_def' by auto

 lemma crel_deterministic: "\<lbrakk> crel f h h' a; crel f h h'' b \<rbrakk> \<Longrightarrow> (a = b) \<and> (h' = h'')"

 unfolding crel_def' by auto

 section {* Elimination rules *}

 text {* For all commands, we define simple elimination rules. *}

 (* FIXME: consumes 1 necessary ?? *)

 subsection {* Elimination rules for basic monadic commands *}

 lemma crelE[consumes 1]:

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

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

   using assms

   by (auto simp add: crel_def bindM_def Let_def prod_case_beta split_def Pair_fst_snd_eq split add: sum.split_asm)

 lemma crelE'[consumes 1]:

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

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

   using assms

   by (elim crelE) auto

 lemma crel_return[consumes 1]:

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

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

   using assms

   unfolding crel_def return_def by simp

 lemma crel_raise[consumes 1]:

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

   obtains "False"

   using assms

   unfolding crel_def raise_def by simp

 lemma crel_if:

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

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

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

   using assms

   unfolding crel_def by auto

 lemma crel_option_case:

   assumes "crel (case x of None \<Rightarrow> n | Some y \<Rightarrow> s y) h h' r"

   obtains "x = None" "crel n h h' r"

         | y where "x = Some y" "crel (s y) h h' r" 

   using assms

   unfolding crel_def by auto

 lemma crel_mapM:

   assumes "crel (mapM f xs) h h' r"

   assumes "\<And>h h'. P f [] h h' []"

   assumes "\<And>h h1 h' x xs y ys. \<lbrakk> crel (f x) h h1 y; crel (mapM f xs) h1 h' ys; P f xs h1 h' ys \<rbrakk> \<Longrightarrow> P f (x#xs) h h' (y#ys)"

   shows "P f xs h h' r"

 using assms(1)

 proof (induct xs arbitrary: h h' r)

   case Nil  with assms(2) show ?case

     by (auto elim: crel_return)

 next

   case (Cons x xs)

   from Cons(2) obtain h1 y ys where crel_f: "crel (f x) h h1 y"

     and crel_mapM: "crel (mapM f xs) h1 h' ys"

     and r_def: "r = y#ys"

     unfolding mapM.simps

     by (auto elim!: crelE crel_return)

   from Cons(1)[OF crel_mapM] crel_mapM crel_f assms(3) r_def

   show ?case by auto

 qed

 lemma crel_heap:

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

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

   using assms

   unfolding heap_def crel_def apfst_def split_def prod_fun_def by simp_all

 subsection {* Elimination rules for array commands *}

 lemma crel_length:

   assumes "crel (length a) h h' r"

   obtains "h = h'" "r = Heap.length a h'"

   using assms

   unfolding length_def

   by (elim crel_heap) simp

 (* Strong version of the lemma for this operation is missing *) 

 lemma crel_new_weak:

   assumes "crel (Array.new n v) h h' r"

   obtains "get_array r h' = List.replicate n v"

   using assms unfolding  Array.new_def

   by (elim crel_heap) (auto simp:Heap.array_def Let_def split_def)

 lemma crel_nth[consumes 1]:

   assumes "crel (nth a i) h h' r"

   obtains "r = (get_array a h) ! i" "h = h'" "i < Heap.length a h"

   using assms

   unfolding nth_def

   by (auto elim!: crelE crel_if crel_raise crel_length crel_heap)

 lemma crel_upd[consumes 1]:

   assumes "crel (upd i v a) h h' r"

   obtains "r = a" "h' = Heap.upd a i v h"

   using assms

   unfolding upd_def

   by (elim crelE crel_if crel_return crel_raise

     crel_length crel_heap) auto

 (* Strong version of the lemma for this operation is missing *)

 lemma crel_of_list_weak:

   assumes "crel (Array.of_list xs) h h' r"

   obtains "get_array r h' = xs"

   using assms

   unfolding of_list_def 

   by (elim crel_heap) (simp add:get_array_init_array_list)

 lemma crel_map_entry:

   assumes "crel (Array.map_entry i f a) h h' r"

   obtains "r = a" "h' = Heap.upd a i (f (get_array a h ! i)) h"

   using assms

   unfolding Array.map_entry_def

   by (elim crelE crel_upd crel_nth) auto

 lemma crel_swap:

   assumes "crel (Array.swap i x a) h h' r"

   obtains "r = get_array a h ! i" "h' = Heap.upd a i x h"

   using assms

   unfolding Array.swap_def

   by (elim crelE crel_upd crel_nth crel_return) auto

 (* Strong version of the lemma for this operation is missing *)

 lemma crel_make_weak:

   assumes "crel (Array.make n f) h h' r"

   obtains "i < n \<Longrightarrow> get_array r h' ! i = f i"

   using assms

   unfolding Array.make_def

   by (elim crel_of_list_weak) auto

 lemma upt_conv_Cons':

   assumes "Suc a \<le> b"

   shows "[b - Suc a..<b] = (b - Suc a)#[b - a..<b]"

 proof -

   from assms have l: "b - Suc a < b" by arith

   from assms have "Suc (b - Suc a) = b - a" by arith

   with l show ?thesis by (simp add: upt_conv_Cons)

 qed

 lemma crel_mapM_nth:

   assumes

   "crel (mapM (Array.nth a) [Heap.length a h - n..<Heap.length a h]) h h' xs"

   assumes "n \<le> Heap.length a h"

   shows "h = h' \<and> xs = drop (Heap.length a h - n) (get_array a h)"

 using assms

 proof (induct n arbitrary: xs h h')

   case 0 thus ?case

     by (auto elim!: crel_return simp add: Heap.length_def)

 next

   case (Suc n)

   from Suc(3) have "[Heap.length a h - Suc n..<Heap.length a h] = (Heap.length a h - Suc n)#[Heap.length a h - n..<Heap.length a h]"

     by (simp add: upt_conv_Cons')

   with Suc(2) obtain r where

     crel_mapM: "crel (mapM (Array.nth a) [Heap.length a h - n..<Heap.length a h]) h h' r"

     and xs_def: "xs = get_array a h ! (Heap.length a h - Suc n) # r"

     by (auto elim!: crelE crel_nth crel_return)

   from Suc(3) have "Heap.length a h - n = Suc (Heap.length a h - Suc n)" 

     by arith

   with Suc.hyps[OF crel_mapM] xs_def show ?case

     unfolding Heap.length_def

     by (auto simp add: nth_drop')

 qed

 lemma crel_freeze:

   assumes "crel (Array.freeze a) h h' xs"

   obtains "h = h'" "xs = get_array a h"

 proof 

   from assms have "crel (mapM (Array.nth a) [0..<Heap.length a h]) h h' xs"

     unfolding freeze_def

     by (auto elim: crelE crel_length)

   hence "crel (mapM (Array.nth a) [(Heap.length a h - Heap.length a h)..<Heap.length a h]) h h' xs"

     by simp

   from crel_mapM_nth[OF this] show "h = h'" and "xs = get_array a h" by auto

 qed

 lemma crel_mapM_map_entry_remains:

   assumes "crel (mapM (\<lambda>n. map_entry n f a) [Heap.length a h - n..<Heap.length a h]) h h' r"

   assumes "i < Heap.length a h - n"

   shows "get_array a h ! i = get_array a h' ! i"

 using assms

 proof (induct n arbitrary: h h' r)

   case 0

   thus ?case

     by (auto elim: crel_return)

 next

   case (Suc n)

   let ?h1 = "Heap.upd a (Heap.length a h - Suc n) (f (get_array a h ! (Heap.length a h - Suc n))) h"

   from Suc(3) have "[Heap.length a h - Suc n..<Heap.length a h] = (Heap.length a h - Suc n)#[Heap.length a h - n..<Heap.length a h]"

     by (simp add: upt_conv_Cons')

   from Suc(2) this obtain r where

     crel_mapM: "crel (mapM (\<lambda>n. map_entry n f a) [Heap.length a h - n..<Heap.length a h]) ?h1 h' r"

     by (auto simp add: elim!: crelE crel_map_entry crel_return)

   have length_remains: "Heap.length a ?h1 = Heap.length a h" by simp

   from crel_mapM have crel_mapM': "crel (mapM (\<lambda>n. map_entry n f a) [Heap.length a ?h1 - n..<Heap.length a ?h1]) ?h1 h' r"

     by simp

   from Suc(1)[OF this] length_remains Suc(3) show ?case by simp

 qed

 lemma crel_mapM_map_entry_changes:

   assumes "crel (mapM (\<lambda>n. map_entry n f a) [Heap.length a h - n..<Heap.length a h]) h h' r"

   assumes "n \<le> Heap.length a h"  

   assumes "i \<ge> Heap.length a h - n"

   assumes "i < Heap.length a h"

   shows "get_array a h' ! i = f (get_array a h ! i)"

 using assms

 proof (induct n arbitrary: h h' r)

   case 0

   thus ?case

     by (auto elim!: crel_return)

 next

   case (Suc n)

   let ?h1 = "Heap.upd a (Heap.length a h - Suc n) (f (get_array a h ! (Heap.length a h - Suc n))) h"

   from Suc(3) have "[Heap.length a h - Suc n..<Heap.length a h] = (Heap.length a h - Suc n)#[Heap.length a h - n..<Heap.length a h]"

     by (simp add: upt_conv_Cons')

   from Suc(2) this obtain r where

     crel_mapM: "crel (mapM (\<lambda>n. map_entry n f a) [Heap.length a h - n..<Heap.length a h]) ?h1 h' r"

     by (auto simp add: elim!: crelE crel_map_entry crel_return)

   have length_remains: "Heap.length a ?h1 = Heap.length a h" by simp

   from Suc(3) have less: "Heap.length a h - Suc n < Heap.length a h - n" by arith

   from Suc(3) have less2: "Heap.length a h - Suc n < Heap.length a h" by arith

   from Suc(4) length_remains have cases: "i = Heap.length a ?h1 - Suc n \<or> i \<ge> Heap.length a ?h1 - n" by arith

   from crel_mapM have crel_mapM': "crel (mapM (\<lambda>n. map_entry n f a) [Heap.length a ?h1 - n..<Heap.length a ?h1]) ?h1 h' r"

     by simp

   from Suc(1)[OF this] cases Suc(3) Suc(5) length_remains

     crel_mapM_map_entry_remains[OF this, of "Heap.length a h - Suc n", symmetric] less less2

   show ?case

     by (auto simp add: nth_list_update_eq Heap.length_def)

 qed

 lemma crel_mapM_map_entry_length:

   assumes "crel (mapM (\<lambda>n. map_entry n f a) [Heap.length a h - n..<Heap.length a h]) h h' r"

   assumes "n \<le> Heap.length a h"

   shows "Heap.length a h' = Heap.length a h"

 using assms

 proof (induct n arbitrary: h h' r)

   case 0

   thus ?case by (auto elim!: crel_return)

 next

   case (Suc n)

   let ?h1 = "Heap.upd a (Heap.length a h - Suc n) (f (get_array a h ! (Heap.length a h - Suc n))) h"

   from Suc(3) have "[Heap.length a h - Suc n..<Heap.length a h] = (Heap.length a h - Suc n)#[Heap.length a h - n..<Heap.length a h]"

     by (simp add: upt_conv_Cons')

   from Suc(2) this obtain r where 

     crel_mapM: "crel (mapM (\<lambda>n. map_entry n f a) [Heap.length a h - n..<Heap.length a h]) ?h1 h' r"

     by (auto elim!: crelE crel_map_entry crel_return)

   have length_remains: "Heap.length a ?h1 = Heap.length a h" by simp

   from crel_mapM have crel_mapM': "crel (mapM (\<lambda>n. map_entry n f a) [Heap.length a ?h1 - n..<Heap.length a ?h1]) ?h1 h' r"

     by simp

   from Suc(1)[OF this] length_remains Suc(3) show ?case by simp

 qed

 lemma crel_mapM_map_entry:

 assumes "crel (mapM (\<lambda>n. map_entry n f a) [0..<Heap.length a h]) h h' r"

   shows "get_array a h' = List.map f (get_array a h)"

 proof -

   from assms have "crel (mapM (\<lambda>n. map_entry n f a) [Heap.length a h - Heap.length a h..<Heap.length a h]) h h' r" by simp

   from crel_mapM_map_entry_length[OF this]

   crel_mapM_map_entry_changes[OF this] show ?thesis

     unfolding Heap.length_def

     by (auto intro: nth_equalityI)

 qed

 lemma crel_map_weak:

   assumes crel_map: "crel (Array.map f a) h h' r"

   obtains "r = a" "get_array a h' = List.map f (get_array a h)"

 proof

   from assms crel_mapM_map_entry show  "get_array a h' = List.map f (get_array a h)"

     unfolding Array.map_def

     by (fastsimp elim!: crelE crel_length crel_return)

   from assms show "r = a"

     unfolding Array.map_def

     by (elim crelE crel_return)

 qed

 subsection {* Elimination rules for reference commands *}

 (* TODO:

 maybe introduce a new predicate "extends h' h x"

 which means h' extends h with a new reference x.

 Then crel_new: would be

 assumes "crel (Ref.new v) h h' x"

 obtains "get_ref x h' = v"

 and "extends h' h x"

 and we would need further rules for extends:

 extends h' h x \<Longrightarrow> \<not> ref_present x h

 extends h' h x \<Longrightarrow>  ref_present x h'

 extends h' h x \<Longrightarrow> ref_present y h \<Longrightarrow> ref_present y h'

 extends h' h x \<Longrightarrow> ref_present y h \<Longrightarrow> get_ref y h = get_ref y h'

 extends h' h x \<Longrightarrow> lim h' = Suc (lim h)

 *)

 lemma crel_Ref_new:

   assumes "crel (Ref.new v) h h' x"

   obtains "get_ref x h' = v"

   and "\<not> ref_present x h"

   and "ref_present x h'"

   and "\<forall>y. ref_present y h \<longrightarrow> get_ref y h = get_ref y h'"

  (* and "lim h' = Suc (lim h)" *)

   and "\<forall>y. ref_present y h \<longrightarrow> ref_present y h'"

   using assms

   unfolding Ref.new_def

   apply (elim crel_heap)

   unfolding Heap.ref_def

   apply (simp add: Let_def)

   unfolding Heap.new_ref_def

   apply (simp add: Let_def)

   unfolding ref_present_def

   apply auto

   unfolding get_ref_def set_ref_def

   apply auto

   done

 lemma crel_lookup:

   assumes "crel (!ref) h h' r"

   obtains "h = h'" "r = get_ref ref h"

 using assms

 unfolding Ref.lookup_def

 by (auto elim: crel_heap)

 lemma crel_update:

   assumes "crel (ref := v) h h' r"

   obtains "h' = set_ref ref v h" "r = ()"

 using assms

 unfolding Ref.update_def

 by (auto elim: crel_heap)

 lemma crel_change:

   assumes "crel (Ref.change f ref) h h' r"

   obtains "h' = set_ref ref (f (get_ref ref h)) h" "r = f (get_ref ref h)"

 using assms

 unfolding Ref.change_def Let_def

 by (auto elim!: crelE crel_lookup crel_update crel_return)

 subsection {* Elimination rules for the assert command *}

 lemma crel_assert[consumes 1]:

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

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

   using assms

   unfolding assert_def

   by (elim crel_if crel_return crel_raise) auto

 lemma crel_assert_eq: "(\<And>h h' r. crel f h h' r \<Longrightarrow> P r) \<Longrightarrow> f \<guillemotright>= assert P = f"

 unfolding crel_def bindM_def Let_def assert_def

   raise_def return_def prod_case_beta

 apply (cases f)

 apply simp

 apply (simp add: expand_fun_eq split_def)

 apply auto

 apply (case_tac "fst (fun x)")

 apply (simp_all add: Pair_fst_snd_eq)

 apply (erule_tac x="x" in meta_allE)

 apply fastsimp

 done

 section {* Introduction rules *}

 subsection {* Introduction rules for basic monadic commands *}

 lemma crelI:

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

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

   using assms by (simp add: crel_def' bindM_def)

 lemma crelI':

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

   shows "crel (f >> g) h h'' r'"

   using assms by (intro crelI) auto

 lemma crel_returnI:

   shows "crel (return x) h h x"

   unfolding crel_def return_def by simp

 lemma crel_raiseI:

   shows "\<not> (crel (raise x) h h' r)"

   unfolding crel_def raise_def by simp

 lemma crel_ifI:

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

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

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

   using assms

   unfolding crel_def by auto

 lemma crel_option_caseI:

   assumes "\<And>y. x = Some y \<Longrightarrow> crel (s y) h h' r"

   assumes "x = None \<Longrightarrow> crel n h h' r"

   shows "crel (case x of None \<Rightarrow> n | Some y \<Rightarrow> s y) h h' r"

 using assms

 by (auto split: option.split)

 lemma crel_heapI:

   shows "crel (Heap_Monad.heap f) h (snd (f h)) (fst (f h))"

   by (simp add: crel_def apfst_def split_def prod_fun_def)

 lemma crel_heapI':

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

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

   using assms

   by (simp add: crel_def split_def apfst_def prod_fun_def)

 lemma crelI2:

   assumes "\<exists>h' rs'. crel f h h' rs' \<and> (\<exists>h'' rs. crel (g rs') h' h'' rs)"

   shows "\<exists>h'' rs. crel (f\<guillemotright>= g) h h'' rs"

   oops

 lemma crel_ifI2:

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

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

   shows "\<exists> h' r. crel (if c then t else e) h h' r"

   oops

 subsection {* Introduction rules for array commands *}

 lemma crel_lengthI:

   shows "crel (length a) h h (Heap.length a h)"

   unfolding length_def

   by (rule crel_heapI') auto

 (* thm crel_newI for Array.new is missing *)

 lemma crel_nthI:

   assumes "i < Heap.length a h"

   shows "crel (nth a i) h h ((get_array a h) ! i)"

   using assms

   unfolding nth_def

   by (auto intro!: crelI crel_ifI crel_raiseI crel_lengthI crel_heapI')

 lemma crel_updI:

   assumes "i < Heap.length a h"

   shows "crel (upd i v a) h (Heap.upd a i v h) a"

   using assms

   unfolding upd_def

   by (auto intro!: crelI crel_ifI crel_returnI crel_raiseI

     crel_lengthI crel_heapI')

 (* thm crel_of_listI is missing *)

 (* thm crel_map_entryI is missing *)

 (* thm crel_swapI is missing *)

 (* thm crel_makeI is missing *)

 (* thm crel_freezeI is missing *)

 (* thm crel_mapI is missing *)

 subsection {* Introduction rules for reference commands *}

 lemma crel_lookupI:

   shows "crel (!ref) h h (get_ref ref h)"

   unfolding lookup_def by (auto intro!: crel_heapI')

 lemma crel_updateI:

   shows "crel (ref := v) h (set_ref ref v h) ()"

   unfolding update_def by (auto intro!: crel_heapI')

 lemma crel_changeI: 

   shows "crel (Ref.change f ref) h (set_ref ref (f (get_ref ref h)) h) (f (get_ref ref h))"

 unfolding change_def Let_def by (auto intro!: crelI crel_returnI crel_lookupI crel_updateI)

 subsection {* Introduction rules for the assert command *}

 lemma crel_assertI:

   assumes "P x"

   shows "crel (assert P x) h h x"

   using assms

   unfolding assert_def

   by (auto intro!: crel_ifI crel_returnI crel_raiseI)

 section {* Defintion of the noError predicate *}

 text {* We add a simple definitional setting for crel intro rules

   where we only would like to show that the computation does not result in a exception for heap h,

   but we do not care about statements about the resulting heap and return value.*}

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

 where

   "noError c h \<longleftrightarrow> (\<exists>r h'. (Inl r, h') = Heap_Monad.execute c h)"

 lemma noError_def': -- FIXME

   "noError c h \<longleftrightarrow> (\<exists>r h'. Heap_Monad.execute c h = (Inl r, h'))"

   unfolding noError_def apply auto proof -

   fix r h'

   assume "(Inl r, h') = Heap_Monad.execute c h"

   then have "Heap_Monad.execute c h = (Inl r, h')" ..

   then show "\<exists>r h'. Heap_Monad.execute c h = (Inl r, h')" by blast

 qed

 subsection {* Introduction rules for basic monadic commands *}

 lemma noErrorI:

   assumes "noError f h"

   assumes "\<And>h' r. crel f h h' r \<Longrightarrow> noError (g r) h'"

   shows "noError (f \<guillemotright>= g) h"

   using assms

   by (auto simp add: noError_def' crel_def' bindM_def)

 lemma noErrorI':

   assumes "noError f h"

   assumes "\<And>h' r. crel f h h' r \<Longrightarrow> noError g h'"

   shows "noError (f \<guillemotright> g) h"

   using assms

   by (auto simp add: noError_def' crel_def' bindM_def)

 lemma noErrorI2:

 "\<lbrakk>crel f h h' r ; noError f h; noError (g r) h'\<rbrakk>

 \<Longrightarrow> noError (f \<guillemotright>= g) h"

 by (auto simp add: noError_def' crel_def' bindM_def)

 lemma noError_return: 

   shows "noError (return x) h"

   unfolding noError_def return_def

   by auto

 lemma noError_if:

   assumes "c \<Longrightarrow> noError t h" "\<not> c \<Longrightarrow> noError e h"

   shows "noError (if c then t else e) h"

   using assms

   unfolding noError_def

   by auto

 lemma noError_option_case:

   assumes "\<And>y. x = Some y \<Longrightarrow> noError (s y) h"

   assumes "noError n h"

   shows "noError (case x of None \<Rightarrow> n | Some y \<Rightarrow> s y) h"

 using assms

 by (auto split: option.split)

 lemma noError_mapM: 

 assumes "\<forall>x \<in> set xs. noError (f x) h \<and> crel (f x) h h (r x)" 

 shows "noError (mapM f xs) h"

 using assms

 proof (induct xs)

   case Nil

   thus ?case

     unfolding mapM.simps by (intro noError_return)

 next

   case (Cons x xs)

   thus ?case

     unfolding mapM.simps

     by (auto intro: noErrorI2[of "f x"] noErrorI noError_return)

 qed

 lemma noError_heap:

   shows "noError (Heap_Monad.heap f) h"

   by (simp add: noError_def' apfst_def prod_fun_def split_def)

 subsection {* Introduction rules for array commands *}

 lemma noError_length:

   shows "noError (Array.length a) h"

   unfolding length_def

   by (intro noError_heap)

 lemma noError_new:

   shows "noError (Array.new n v) h"

 unfolding Array.new_def by (intro noError_heap)

 lemma noError_upd:

   assumes "i < Heap.length a h"

   shows "noError (Array.upd i v a) h"

   using assms

   unfolding upd_def

   by (auto intro!: noErrorI noError_if noError_return noError_length noError_heap) (auto elim: crel_length)

 lemma noError_nth:

 assumes "i < Heap.length a h"

   shows "noError (Array.nth a i) h"

   using assms

   unfolding nth_def

   by (auto intro!: noErrorI noError_if noError_return noError_length noError_heap) (auto elim: crel_length)

 lemma noError_of_list:

   shows "noError (of_list ls) h"

   unfolding of_list_def by (rule noError_heap)

 lemma noError_map_entry:

   assumes "i < Heap.length a h"

   shows "noError (map_entry i f a) h"

   using assms

   unfolding map_entry_def

   by (auto elim: crel_nth intro!: noErrorI noError_nth noError_upd)

 lemma noError_swap:

   assumes "i < Heap.length a h"

   shows "noError (swap i x a) h"

   using assms

   unfolding swap_def

   by (auto elim: crel_nth intro!: noErrorI noError_return noError_nth noError_upd)

 lemma noError_make:

   shows "noError (make n f) h"

   unfolding make_def

   by (auto intro: noError_of_list)

 (*TODO: move to HeapMonad *)

 lemma mapM_append:

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

   by (induct xs) (simp_all add: monad_simp)

 lemma noError_freeze:

   shows "noError (freeze a) h"

 unfolding freeze_def

 by (auto intro!: noErrorI noError_length noError_mapM[of _ _ _ "\<lambda>x. get_array a h ! x"]

   noError_nth crel_nthI elim: crel_length)

 lemma noError_mapM_map_entry:

   assumes "n \<le> Heap.length a h"

   shows "noError (mapM (\<lambda>n. map_entry n f a) [Heap.length a h - n..<Heap.length a h]) h"

 using assms

 proof (induct n arbitrary: h)

   case 0

   thus ?case by (auto intro: noError_return)

 next

   case (Suc n)

   from Suc.prems have "[Heap.length a h - Suc n..<Heap.length a h] = (Heap.length a h - Suc n)#[Heap.length a h - n..<Heap.length a h]"

     by (simp add: upt_conv_Cons')

   with Suc.hyps[of "(Heap.upd a (Heap.length a h - Suc n) (f (get_array a h ! (Heap.length a h - Suc n))) h)"] Suc.prems show ?case

     by (auto simp add: intro!: noErrorI noError_return noError_map_entry elim: crel_map_entry)

 qed

 lemma noError_map:

   shows "noError (Array.map f a) h"

 using noError_mapM_map_entry[of "Heap.length a h" a h]

 unfolding Array.map_def

 by (auto intro: noErrorI noError_length noError_return elim!: crel_length)

 subsection {* Introduction rules for the reference commands *}

 lemma noError_Ref_new:

   shows "noError (Ref.new v) h"

 unfolding Ref.new_def by (intro noError_heap)

 lemma noError_lookup:

   shows "noError (!ref) h"

   unfolding lookup_def by (intro noError_heap)

 lemma noError_update:

   shows "noError (ref := v) h"

   unfolding update_def by (intro noError_heap)

 lemma noError_change:

   shows "noError (Ref.change f ref) h"

   unfolding Ref.change_def Let_def by (intro noErrorI noError_lookup noError_update noError_return)

 subsection {* Introduction rules for the assert command *}

 lemma noError_assert:

   assumes "P x"

   shows "noError (assert P x) h"

   using assms

   unfolding assert_def

   by (auto intro: noError_if noError_return)

 section {* Cumulative lemmas *}

 lemmas crel_elim_all =

   crelE crelE' crel_return crel_raise crel_if crel_option_case

   crel_length crel_new_weak crel_nth crel_upd crel_of_list_weak crel_map_entry crel_swap crel_make_weak crel_freeze crel_map_weak

   crel_Ref_new crel_lookup crel_update crel_change

   crel_assert

 lemmas crel_intro_all =

   crelI crelI' crel_returnI crel_raiseI crel_ifI crel_option_caseI

   crel_lengthI (* crel_newI *) crel_nthI crel_updI (* crel_of_listI crel_map_entryI crel_swapI crel_makeI crel_freezeI crel_mapI *)

   (* crel_Ref_newI *) crel_lookupI crel_updateI crel_changeI

   crel_assert

 lemmas noError_intro_all = 

   noErrorI noErrorI' noError_return noError_if noError_option_case

   noError_length noError_new noError_nth noError_upd noError_of_list noError_map_entry noError_swap noError_make noError_freeze noError_map

   noError_Ref_new noError_lookup noError_update noError_change

   noError_assert

 end