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